Research ArticleRepresenting and Counting the Subgroups of the Group Z

119898times Z

119899

Mario Hampejs1 Nicki Holighaus2 Laacuteszloacute Toacuteth34 and Christoph Wiesmeyr1

1 NuHAG Faculty of Mathematics University of Vienna Oskar Morgenstern Platz 1 1090 Vienna Austria2 Acoustics Research Institute Austrian Academy of Sciences Wohllebengasse 12-14 1040 Vienna Austria3 Department of Mathematics University of Pecs Ifjusag Utja 6 Pecs 7624 Hungary4 Institute of Mathematics University of Natural Resources and Life Sciences Gregor-Mendel-Straszlige 33 1180 Vienna Austria

Correspondence should be addressed to Laszlo Toth ltothgammattkptehu

Received 28 July 2014 Accepted 14 September 2014 Published 16 October 2014

Academic Editor Andrej Dujella

Copyright copy 2014 Mario Hampejs et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We deduce a simple representation and the invariant factor decompositions of the subgroups of the group Z119898times Z119899 where119898 and

119899 are arbitrary positive integers We obtain formulas for the total number of subgroups and the number of subgroups of a givenorder

1 Introduction

LetZ119898be the group of residue classesmodulo119898 and consider

the direct product 119866 = Z119898

times Z119899 where 119898 and 119899 are

arbitrary positive integersThis paper aims to deduce a simplerepresentation and the invariant factor decompositions ofthe subgroups of the group 119866 As consequences we deriveformulas for the number of certain types of subgroups of 119866including the total number 119904(119898 119899) of its subgroups and thenumber 119904

119896(119898 119899) of its subgroups of order 119896 (119896 | 119898119899)

Subgroups of Z times Z (sublattices of the two-dimensionalinteger lattice) and associated counting functions were con-sidered by several authors in pure and appliedmathematics Itis known for example that the number of subgroups of index119899 in Z times Z is 120590(119899) the sum of the (positive) divisors of 119899See for example [1 2] [3 item A001615] Although featuresof the subgroups of 119866 not only are interesting by their ownbut also have applications one of them is described below itseems that a synthesis on subgroups of 119866 cannot be found inthe literature

In the case 119898 = 119899 the subgroups of Z119899times Z119899play

an important role in numerical harmonic analysis morespecifically in the field of applied time-frequency analysisTime-frequency analysis attempts to investigate functionbehavior via a phase space representation given by theshort-time Fourier transform [4] The short-time Fouriercoefficients of a function 119891 are given by inner products

with translated modulations (or time-frequency shifts) ofa prototype function 119892 assumed to be well localized inphase space for example a Gaussian In applications thephase space corresponding to discrete finite functions (orvectors) belonging to C119899 is exactly Z

119899times Z119899 Concerned with

the question of reconstruction from samples of short-timeFourier transforms it has been found that when samplingon lattices that is subgroups of Z

119899times Z119899 the associated

analysis and reconstruction operators are particularly richin structure which in turn can be exploited for efficientimplementation (cf [5ndash7] and references therein) It is ofparticular interest to find subgroups in a certain range ofcardinality therefore a complete characterization of thesegroups helps choose the best one for the desired application

We recall that a finite Abelian group of order gt 1 has rank119903 if it is isomorphic toZ

1198991

timessdot sdot sdottimesZ119899119903

where 1198991 119899

119903isin N1

and 119899119895

| 119899119895+1

(1 le 119895 le 119903 minus 1) which is the invariantfactor decomposition of the given group Here the number 119903is uniquely determined and represents the minimal numberof generators of the group For general accounts on finiteAbelian groups see for example [8 9]

It is known that for every finite Abelian group theproblem of counting all subgroups and the subgroups of agiven order reduces to 119901-groups which follows from theproperties of the subgroup lattice of the group (see [10 11])In particular for 119866 = Z

119898times Z119899 this can be formulated as

follows Assume that gcd(119898 119899) gt 1 Then 119866 is an Abelian

Hindawi Publishing CorporationJournal of NumbersVolume 2014 Article ID 491428 6 pageshttpdxdoiorg1011552014491428

2 Journal of Numbers

group of rank two since 119866 ≃ Z119906times ZV where 119906 = gcd(119898 119899)

and V = lcm(119898 119899) Let 119906 = 1199011198861

1sdot sdot sdot 119901119886119903

119903and V = 119901

1198871

1sdot sdot sdot 119901119887119903

119903be

the prime power factorizations of 119906 and V respectively where0 le 119886119895le 119887119895(1 le 119895 le 119903) Then

119904 (119898 119899) =

119903

prod

119895=1

119904 (119901

119886119895

119895 119901

119887119895

119895) (1)

119904119896(119898 119899) =

119903

prod

119895=1

119904119896119895

(119901

119886119895

119895 119901

119887119895

119895) (2)

where 119896 = 1198961sdot sdot sdot 119896119903and 119896

119895= 119901

119888119895

119895with some exponents 0 le

119888119895le 119886119895+ 119887119895(1 le 119895 le 119903)

Now consider the 119901-group Z119901119886 times Z119901119887 where 0 le 119886 le 119887

This is of rank two for 1 le 119886 le 119887 One has the simple explicitformulae

119904 (119901119886

119901119887

)

=

(119887 minus 119886 + 1) 119901119886+2

minus (119887 minus 119886 minus 1) 119901119886+1

minus (119886 + 119887 + 3) 119901 + (119886 + 119887 + 1)

(119901 minus 1)2

(3)

119904119901119888 (119901119886

119901119887

) =

119901119888+1

minus 1

119901 minus 1

119888 le 119886 le 119887

119901119886+1

minus 1

119901 minus 1

119886 le 119888 le 119887

119901119886+119887minus119888+1

minus 1

119901 minus 1

119886 le 119887 le 119888 le 119886 + 119887

(4)

Formula (3) was derived by Calugareanu [12 Section 4]and recently by Petrillo [13 Proposition 2] using Goursatrsquoslemma for groups Tarnauceanu [14 Proposition 29] and [15Theorem 33] deduced (3) and (4) by a method based onproperties of certain attached matrices

Therefore 119904(119898 119899) and 119904119896(119898 119899) can be computed using (1)

(3) and (2) (4) respectively We deduce other formulas for119904(119898 119899) and 119904

119896(119898 119899) (Theorems 3 and 4) which generalize

(3) and (4) and put them in more compact forms These areconsequences of a simple representation of the subgroups of119866 = Z

119898timesZ119899 given inTheorem 1 This representation might

be known but the only source we could find is the paper[5] where only a special case is treated in a different formMore exactly in [5 Lemma 41] a representation for latticesin Z119899times Z119899of redundancy 2 that is subgroups of Z

119899times Z119899

having index 1198992 is given using matrices in Hermite normalform Theorem 2 gives the invariant factor decompositionsof the subgroups of 119866 We also consider the number ofcyclic subgroups of Z

119898times Z119899(Theorem 5) and the number

of subgroups of a given exponent in Z119899times Z119899(Theorem 8)

Our approach is elementary using only simple group-theoretic and number-theoretic arguments The proofs aregiven in Section 4

Throughout the paper we use the following notationsN = 1 2 N

0= 0 1 2 120591(119899) and 120590(119899) are the

number and the sum respectively of the positive divisors of119899 120595(119899) = 119899prod

119901|119899(1 + 1119901) is the Dedekind function 120596(119899)

stands for the number of distinct prime factors of 119899 120583 is theMobius function 120601 denotes Eulerrsquos totient function and 120577 isthe Riemann zeta function

s

b

a

1

1

0

0

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

Figure 1

2 Subgroups of Z119898times Z119899

The subgroups ofZ119898timesZ119899can be identified and visualized in

the plane with sublattices of the lattice Z119898times Z119899 Every two-

dimensional sublattice is generated by two basis vectors Forexample Figure 1 shows the subgroup ofZ

12timesZ12having the

basis vectors (3 0) and (1 2)This suggests the following representation of the sub-

groups

Theorem 1 For every119898 119899 isin N let

119868119898119899

= (119886 119887 119905) isin N2 times N0 119886 | 119898 119887 | 119899

0 le 119905 le gcd(119886 119899119887

) minus 1

(5)

and for (119886 119887 119905) isin 119868119898119899

define

119867119886119887119905

= (119894119886 +

119895119905119886

gcd (119886 119899119887)

119895119887)

0 le 119894 le

119898

119886

minus 1 0 le 119895 le

119899

119887

minus 1

(6)

Then 119867119886119887119905

is a subgroup of order 119898119899119886119887 of Z119898times Z119899and

the map (119886 119887 119905) 997891rarr 119867119886119887119905

is a bijection between the set 119868119898119899

andthe set of subgroups of Z

119898times Z119899

Note that for the subgroup 119867119886119887119905

the basis vectorsmentioned above are (119886 0) and (119904 119887) where

119904 =

119905119886

gcd (119886 119899119887) (7)

This notation for 119904 will be used also in the rest of thepaper Note also that in the case 119886 = 119898 119887 = 119899 the area of

Journal of Numbers 3

the parallelogram spanned by the basis vectors is 119886119887 exactlythe index of119867

119886119887119905

We say that a subgroup119867 = 119867119886119887119905

is a subproduct ofZ119898times

Z119899if 119867 = 119867

1times 1198672 where 119867

1and 119867

2are subgroups of Z

119898

and Z119899 respectively

Theorem 2 (i) The invariant factor decomposition of thesubgroup119867

119886119887119905is given by

119867119886119887119905

≃ Z120572times Z120573 (8)

where

120572 = (

119898

119886

119899

119887

119899119904

119886119887

) 120573 =

119898119899

a119887120572(9)

satisfying 120572 | 120573(ii) The exponent of the subgroup119867

119886119887119905is 120573

(iii) The subgroup119867119886119887119905

is cyclic if and only if 120572 = 1(iv) The subgroup119867

119886119887119905is a subproduct if and only if 119905 = 0

and 1198671198861198870

= Z119898119886

times Z119899119887

Here 1198671198861198870

is cyclic if and only ifgcd (119898119886 119899119887) = 1

For example for the subgroup represented by Figure 1 onehas 119898 = 119899 = 12 119886 = 3 119887 = 2 119904 = 1 120572 = 2 and 120573 = 12 andthis subgroup is isomorphic to Z

2times Z12 It is not cyclic and

is not a subproductAccording toTheorem 1 the number 119904(119898 119899) of subgroups

of Z119898times Z119899can be obtained by counting the elements of the

set 119868119898119899

We deduce the following

Theorem 3 For every119898 119899 isin N 119904(119898 119899) is given by

119904 (119898 119899) = sum

119886|119898 119887|119899

gcd (119886 119887) (10)

= sum

119889|gcd(119898119899)120601 (119889) 120591 (

119898

119889

) 120591 (

119899

119889

) (11)

= sum

119889|gcd(119898119899)119889120591 (

119898119899

1198892) (12)

Formula (10) is a special case of a formula representingthe number of all subgroups of a class of groups formed ascyclic extensions of cyclic groups deduced by Calhoun [16]and having a laborious proof Note that formula (10) is givenwithout proof in [3 item A054584]

Note also that the function (119898 119899) 997891rarr 119904(119898 119899) is represent-ing a multiplicative arithmetic function of two variables thatis 119904(119898119898

1015840

1198991198991015840

) = 119904(119898 119899)119904(1198981015840

1198991015840

) holds for any 119898 1198991198981015840

1198991015840

isin

N such that gcd(1198981198991198981015840

1198991015840

) = 1 This property which is inconcordance with (1) is a direct consequence of formula (10)See Section 5

Let119873(119886 119887 119888) denote the number of solutions (119909 119910 119911 119905) isinN4 of the system of equations 119909119910 = 119886 119911119905 = 119887 119909119911 = 119888

Theorem 4 For every 119896119898 119899 isin N such that 119896 | 119898119899

119904119896(119898 119899) = sum

119886|119898119887|119899

119898119887119886=119896

gcd (119886 119887)(13)

= sum

119889|gcd(119896119898)119890| gcd (119896119899)119896|119889119890

120601(

119889119890

119896

)

(14)

= sum

119889|gcd(119898119899119896)120601 (119889)119873(

119898

119889

119899

119889

119896

119889

) (15)

The identities (3) and (4) can be easily deduced from eachof the identities given inTheorems 3 and 4 respectively

Theorem 5 Let119898 119899 isin N(i) The number 119888(119898 119899) of cyclic subgroups of Z

119898times Z119899is

given by

119888 (119898 119899) = sum

119886|119898119887|119899

gcd(119898119886119899119887)=1

gcd (119886 119887)(16)

= sum

119886|119898119887|119899

120601 (gcd (119886 119887)) (17)

= sum

119889|gcd(119898119899)(120583 lowast 120601) (119889) 120591 (

119898

119889

) 120591 (

119899

119889

) (18)

= sum

119889|gcd(119898119899)120601 (119889) 120591 (

119898119899

1198892) (19)

(ii)The number of subproducts ofZ119898timesZ119899is 120591(119898)120591(119899) and

the number of its cyclic subproducts is 120591(119898119899)

Formula (17) as a special case of an identity valid forarbitrary finite Abelian groups was derived by the thirdauthor [17 18] using different arguments The function(119898 119899) 997891rarr 119888(119898 119899) is also multiplicative

3 Subgroups of Z119899times Z119899

In the case119898 = 119899 which is of special interest in applicationsthe results given in the previous section can be easily usedWe point out that 119899 997891rarr 119904(119899) = 119904(119899 119899) and 119899 997891rarr 119888(119899) =

119888(119899 119899) are multiplicative arithmetic functions of a singlevariable (sequences [3 items A060724A060648]) They canbe written in the form of Dirichlet convolutions as shown bythe next corollaries

Corollary 6 For every 119899 isin N

119904 (119899) = sum

119889119890=119899

120601 (119889) 1205912

(119890)

= sum

119889119890=119899

119889 120591 (1198902

)

(20)

4 Journal of Numbers

Corollary 7 For every 119899 isin N

119888 (119899) = sum

119889119890=119899

1198892120596(119890)

(21)

= sum

119889119890=119899

120601 (119889) 120591 (1198902

) (22)

Further convolutional representations can also be givenfor example

119904 (119899) = sum

119889119890=119899

120591 (119889) 120595 (119890) 119888 (119899) = sum

119889|119899

120595 (119889) (23)

all of these follow from the Dirichlet-series representationsinfin

sum

119899=1

119904 (119899)

119899119911

=

1205773

(119911) 120577 (119911 minus 1)

120577 (2119911)

(24)

infin

sum

119899=1

119888 (119899)

119899119911

=

1205772

(119911) 120577 (119911 minus 1)

120577 (2119911)

(25)

valid for 119911 isin CR(119911) gt 2Observe that

119904 (119899) = sum

119889|119899

119888 (119889) (119899 isin N) (26)

which is a simple consequence of (23) or of (24) and (25) Italso follows from the next result

Theorem 8 For every 119899 120575 isin N with 120575 | 119899 the number ofsubgroups of exponent 120575 ofZ

119899timesZ119899equals the number of cyclic

subgroups of Z120575times Z120575

4 Proofs

Proof of Theorem 1 Let 119867 be a subgroup of 119866 = Z119898times Z119899

Consider the natural projection 1205872

119866 rarr Z119899given by

1205872(119909 119910) = 119910 Then 120587

2(119867) is a subgroup of Z

119899and there is

a unique divisor 119887 of 119899 such that 1205872(119867) = ⟨119887⟩ = 119895119887 0 le 119895 le

119899119887 minus 1 Let 119904 ge 0 be minimal such that (119904 119887) isin 119867Furthermore consider the natural inclusion 120580

1 Z119898

rarr

119866 given by 1205801(119909) = (119909 0)Then 120580minus1

1(119867) is a subgroup ofZ

119898and

there exists a unique divisor 119886 of119898 such that 120580minus11(119867) = ⟨119886⟩

We show that 119867 = (119894119886 + 119895119904 119895119887) 119894 119895 isin Z Indeed forevery 119894 119895 isin Z (119894119886+119895119904 119895119887) = 119894(119886 0)+119895(119904 119887) isin 119867 On the otherhand for every (119906 V) isin 119867 one has V isin 120587

2(119867) and hence there

is 119895 isin Z such that V = 119895119887 We obtain (119906 minus 119895119904 0) = (119906 V) minus119895(119904 119887) isin 119867 119906minus119895119904 isin 120580

minus1

1(119867) and there is 119894 isin Zwith 119906minus119895119904 = 119894119886

Here a necessary condition is that (119904119899119887 0) isin 119867 (obtainedfor 119894 = 0 and 119895 = 119899119887) that is 119886 | 119904119899b equivalent to119886gcd(119886 119899119887) | 119904 Clearly if this is verified then for the aboverepresentation of 119867 it is enough to take the values 0 le 119894 le

119898119886 minus 1 and 0 le 119895 le 119899119887 minus 1Also dividing 119904 by 119886 we have 119904 = 119886119902 + 119903 with 0 le 119903 lt 119886

and (119903 119887) = (119904 119887) minus 119902(119886 0) isin 119867 showing that 119904 lt 119886 byits minimality Hence 119904 = 119905119886gcd(119886 119899119887) with 0 le 119905 le

gcd(119886 119899119887) minus 1 Thus we obtain the given representationConversely every (119886 119887 119905) isin 119868

119898119899generates a subgroup

119867119886119887119905

of order 119898119899(119886119887) of Z119898

times Z119899and the proof is

complete

Proof of Theorem 2 (i)-(ii) We first determine the exponentof the subgroup 119867

119886119887119905 119867119886119887119905

is generated by (119886 0) and (119904 119887)hence its exponent is the least commonmultiple of the ordersof these two elements The order of (119886 0) is119898119886 To computethe order of (119904 119887) note that119898 | 119903119904 if and only if119898gcd(119898 119904) |

119903Thus the order of (119904 119887) is lcm(119898gcd(119898 119904) 119899119887)We deducethat the exponent of119867

119886119887119905is

lcm(

119898

119886

119898

gcd (119898 119904)

119899

119887

)

= lcm(

119898119899

119899119886

119898119899

119899gcd (119898 119904)

119898119899

119898119887

)

=

119898119899

gcd (119899119886 119899119898 119899119904 119898119887)

=

119898119899

gcd (119898119887 119899119886 119899119904)

= 120573

(27)

For every finite Abelian group the rank of a nontrivialsubgroup is at most the rank of the groupTherefore the rankof 119867119886119887t is 1 or 2 That is 119867

119886119887119905≃ Z119860times Z119861with certain

119860 119861 isin N such that 119860 | 119861 Here the exponent of 119867119886119887119905

equalsthat of Z

119860times Z119861 which is lcm(119860 119861) = 119861 Using (ii) already

proved we deduce that 119861 = 120573 Since the order of 119867119886119887119905

is119860119861 = 119898119899(119886119887) we have 119860 = 119898119899(119886119887120573) = 120572

(iii) According to (i) 119867119886119887119905

≃ Z120572timesZ120573 where 120572 | 120573 Hence

119867119886119887119905

is cyclic if and only if 120572 = 1(iv) The subgroups of Z

119898are of form 119894119886 0 le 119894 le 119898119886 minus

1 where 119886 | 119898 and the properties follow from (6) and (iii)

Proof of Theorem 3 By its definition the number of elementsof the set 119868

119898119899is

sum

119886|119898119887|119899

sum

0le119905legcd(119886119899119887)minus11

= sum

119886|119898119887|119899

gcd(119886 119899119887

) = sum

119886|119898119887|119899

gcd (119886 119887) (28)

representing 119904(119898 119899) This is formula (10)To obtain formula (11) apply the Gauss formula 119899 =

sum119889|119899

120601(119889) (119899 isin N) by writing the following

119904 (119898 119899) = sum

119886|119898119887|119899

sum

119889|gcd(119886119887)120601 (119889) = sum

119886119909=119898

119887119910=119899

sum

119889119894=119886

119889119895=119887

120601 (119889) = sum

119889119894119909=119898

119889119895119910=119899

120601 (119889)

= sum

119889119906=119898

119889V=119899

120601 (119889) sum

119894119909=119906

119895119910=V

1 = sum

119889119906=119898

119889V=119899

120601 (119889) 120591 (119906) 120591 (V)

= sum

119889|gcd(119898119899)120601 (119889) 120591 (

119898

119889

) 120591 (

119899

119889

)

(29)

Journal of Numbers 5

Now (12) follows from (11) by the Busche-Ramanujanidentity (cf [19 Chapter 1])

120591 (119898) 120591 (119899) = sum

119889|gcd(119898119899)120591 (

119898119899

1198892) (119898 119899 isin N) (30)

Proof of Theorem 4 According toTheorem 1

119904119896(119898 119899) = sum

119886|119898119887|119899

119898119899119886119887=119896

gcd(119886 119899119887

) (31)

giving (13) which can be written by Gaussrsquo formula again as

119904119896(119898 119899) = sum

119886|119898119887|119899

119898119887119886=119896

sum

119888|119886119888|119887

120601 (119888) = sum

119888119894119909=119898

119888119895119910=119899

119888119895119909=119896

120601 (119888)

= sum

119889119894=119898

119890119910=119899

sum

119888119909=119889

119888119895=119890

119888119895119909=119896

120601 (119888)

(32)

where in the inner sum one has 119888 = 119889119890119896 and obtains (14)Now to get (15) write (32) as

119904119896(119898 119899) = sum

119888119906=119898

119888V=119899119888119908=119896

120601 (119888) sum

119894119909=119906

119895119910=V119895119909=119908

1 = sum

119888119906=119898

119888V=119899119888119908=119896

120601 (119888)119873 (119906 V 119908)

= sum

119888|gcd(119898119899119896)120601 (119888)119873(

119898

119888

119899

119888

119896

119888

)

(33)

and the proof is complete

Proof of Theorem 5 (i) According to Theorems 1 and 2(iii)and using that sum

119889|119899120583(119889) = 1 or 0 and according to 119899 = 1 or

119899 gt 1

119888 (119898 119899) = sum

119886|119898119887|119899

sum

1le119904le119886

119886119887|119899119904

gcd(119898119886119899119887119899119904119886119887)=1

1 = sum

119886119909=119898

119887119910=119899

sum

1le119904le119886

119886119903=119910119904

gcd(119909119910119903)=1

1

= sum

119886119909=119898

119887119910=119899

sum

1le119904le119886

119886119903=119910119904

sum

119890|gcd(119909119910119903)

120583 (119890) = sum

119886119890119894=119898

119887119890119895=119899

120583 (119890) sum

1le119904le119886

119886gcd(119886119895)|119904

1

(34)

where the inner sum is gcd(119886 119895) Hence

119888 (119898 119899) = sum

119886119890119894=119898

119887119890119895=119899

120583 (119890) gcd (119886 119895) (35)

Now regrouping the terms according to 119890119894 = 119911 and 119887119890 = 119905

we obtain

119888 (119898 119899) = sum

119886119911=119898

119895119905=119899

gcd (119886 119895) sum119890119894=119911

119887119890=119905

120583 (119890) = sum

119886119911=119898

119895119905=119899

gcd (119886 119895) sum

119890|gcd(119911119905)120583 (119890)

= sum

119886119911=119898

119895119905=119899

gcd(119911119905)=1

gcd (119886 119895)

(36)

which is (16)

The next results follow applying Gaussrsquo formula andthe Busche-Ramanujan formula similar to the proof ofTheorem 3

(ii) For the subproducts 1198671198861198870

the values 119886 | 119898 and 119887 | 119899

can be chosen arbitrary and it follows at once that the numberof subproducts is 120591(119898)120591(119899)Thenumber of cyclic subproductsis

sum

119886|119898

119887|119899

gcd(119898119886119899119887)=1

1 = sum

119886119909=119898

119887119910=119899

gcd(119909119910)=1

1 = sum

119886119909=119898

119887119910=119899

sum

119890|gcd(119909119910)

120583 (119890)

= sum

119890119860=119898

119890119861=119899

120583 (119890) 120591 (119860) 120591 (119861)

= sum

119890|gcd(119898119899)120583 (119890) 120591 (

119898

119890

) 120591 (

119899

119890

) = 120591 (119898119899)

(37)

by the inverse Busche-Ramanujan identity

Proof of Theorem 8 According to Theorem 2(ii) the num-ber of subgroups of exponent 120575 of Z

119899times Z119899is

119864120575(119899) = sum

119886|119899119887|119899

sum

1le119904le119886

119886119887|119899119904

119899gcd(119886119887119904)=120575

1 = sum

119886119909=119899

119887119910=119899

sum

1le119904le119886

119886119903=119910119904

gcd(119886119887119904)=119899120575

1

(38)

Write 119886 = 1198861119899120575 119887 = 119887

1119899120575 and 119904 = 119904

1119899120575 with

gcd(1198861 1198871 1199041) = 1 We deduce similar to the proof of

Theorem 5(i) that

119864120575(119899) = sum

119890119894119909=120575

119890119895119910=120575

120583 (119890) gcd (119894 119910) (39)

which is exactly 119888(120575 120575) = 119888(120575) (cf (35))

5 Further Remarks

(1) As mentioned in Section 2 the functions (119898 119899) 997891rarr

119904(119898 119899) and (119898 119899) 997891rarr 119888(119898 119899) are multiplicative func-tions of two variablesThis follows easily from formu-lae (10) and (17) respectively Namely according tothose formulae 119904(119898 119899) and 119888(119898 119899) are two variablesDirichlet convolutions of the functions (119898 119899) 997891rarr

gcd(119898 119899) and (119898 119899) 997891rarr 120601(gcd(119898 119899)) respectivelywith the constant 1 function all multiplicative Sinceconvolution preserves the multiplicativity we deducethat 119904(119898 119899) and 119888(119898 119899) are alsomultiplicative See [17Section 2] for details

(2) Asymptotic formulas with sharp error terms for thesums sum

119898119899le119909119904(119898 119899) and sum

119898119899le119909119888(119898 119899) were given in

the paper [20](3) For any finite groups 119860 and 119861 a subgroup 119862 of 119860 times 119861

is cyclic if and only if 120580minus11(119862) and 120580

minus1

2(119862) have coprime

orders where 1205801and 1205802are the natural inclusions ([21

Theorem 42]) In the case 119860 = Z119898 119861 = Z

119899 and

119862 = 119867119886119887119905

one has 120580minus11(119862) = 119898119886 and 120580minus1

2(119862) =

gcd(119899119887 119899119904119886119887) and the characterization of the cyclicsubgroups 119867

119886119887119905given in Theorem 2(iii) can be

6 Journal of Numbers

obtained also in this way It turns out that regardingthe sublattice 119867

119886119887119905is cyclic if and only if the

numbers of points on the horizontal and vertical axesrespectively are relatively prime Note that in the case119898 = 119899 the above condition reads 119899gcd(119886 119887 119904) = 119886119887Thus it is necessary that 119899 | 119886119887 The subgroup onFigure 1 is not cyclic

(4) Note also the next formula for the number of cyclicsubgroups of Z

119899times Z119899 derived in [22 Example 2]

119888 (119899) = sum

lcm(119889119890)=119899gcd (119889 119890) (119899 isin N) (40)

where the sum is over all ordered pairs (119889 119890) such thatlcm(119889 119890) = 119899 For a short direct proof of (40) write119889 = ℓ119886 119890 = ℓ119887 with gcd(119886 119887) = 1 Then gcd(119889 119890) = ℓlcm(119889 119890) = ℓ119886119887 and obtain

sum

lcm(119889119890)=119899gcd (119889 119890)

= sum

ℓ119886119887=119899

gcd(119886119887)=1

ℓ = sum

ℓ119896=119899

ℓ sum

119886119887=119896

gcd(119886119887)=1

1 = sum

ℓ119896=119899

ℓ2120596(119896)

= 119888 (119899)

(41)

according to (21)(5) Every subgroup119870 ofZtimesZ has the representation119870 =

(119894119886 + 119895119904 119895119887) 119894 119895 isin Z where 0 le 119904 le 119886 and 0 le 119887

are unique integers This follows like in the proof ofTheorem 1 Furthermore in the case 119886 119887 ge 1 0 le 119904 le

119886minus 1 the index of119870 is 119886119887 and one obtains at once thatthe number of subgroups 119870 having index 119899 (119899 isin N)is sum119886119887=119899

sum0le119904le119886minus1

1 = sum119886119887=119899

119886 = 120590(119899) mentioned inSection 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Nicki Holighaus was partially supported by the AustrianScience Fund (FWF) START-Project FLAME (Y551-N13)Laszlo Toth gratefully acknowledges support from the Aus-trian Science Fund (FWF) under Project no M1376-N18Christoph Wiesmeyr was partially supported by EU FETOpen Grant UNLocX (255931)

References

[1] M J Grady ldquoA group theoretic approach to a famous partitionformulardquo The American Mathematical Monthly vol 112 no 7pp 645ndash651 2005

[2] Y M Zou ldquoGaussian binomials and the number of sublatticesrdquoActa Crystallographica vol 62 no 5 pp 409ndash410 2006

[3] ldquoThe On-Line Encyclopedia of Integer Sequencesrdquo httpoeisorg

[4] K Grochenig Foundations of Time-Frequency Analysis Appliedand Numerical Harmonic Analysis Birkhauser Boston MassUSA 2001

[5] G Kutyniok and T Strohmer ldquoWilson bases for general time-frequency latticesrdquo SIAM Journal onMathematical Analysis vol37 no 3 pp 685ndash711 (electronic) 2005

[6] A J van Leest Non-separable Gabor schemes their design andimplementation [PhD thesis] Technische Universiteit Eind-hoven Eindhoven The Netherlands 2001

[7] T Strohmer ldquoNumerical algorithms for discrete Gabor expan-sionsrdquo in Gabor Analysis and Algorithms Theory and Applica-tions H G Feichtinger and T Strohmer Eds pp 267ndash294Birkhauser Boston Mass USA 1998

[8] A Machı Groups An Introduction to Ideas and Methods of theTheory of Groups Springer 2012

[9] J J Rotman An Introduction to theTheory of Groups vol 148 ofGraduate Texts in Mathematics Springer New York NY USA4th edition 1995

[10] R Schmidt Subgroup Lattices of Groups deGruyter Expositionsin Mathematics 14 de Gruyter Berlin Germany 1994

[11] M Suzuki ldquoOn the lattice of subgroups of finite groupsrdquoTransactions of the American Mathematical Society vol 70 pp345ndash371 1951

[12] G Calugareanu ldquoThe total number of subgroups of a finiteabelian grouprdquo Scientiae Mathematicae Japonicae vol 60 no1 pp 157ndash167 2004

[13] J Petrillo ldquoCounting subgroups in a direct product of finitecyclic groupsrdquo College Mathematics Journal vol 42 no 3 pp215ndash222 2011

[14] M Tarnauceanu ldquoA new method of proving some classicaltheorems of abelian groupsrdquo Southeast Asian Bulletin of Mathe-matics vol 31 no 6 pp 1191ndash1203 2007

[15] M Tarnauceanu ldquoAn arithmetic method of counting the sub-groups of a finite abelian grouprdquo Bulletin Mathematiques de laSociete des SciencesMathematiques de Roumanie vol 53 no 101pp 373ndash386 2010

[16] W C Calhoun ldquoCounting the subgroups of some finite groupsrdquoThe American Mathematical Monthly vol 94 no 1 pp 54ndash591987

[17] L Toth ldquoMenons identity and arithmetical sums representingfunctions of several variablesrdquoRendiconti del SeminarioMatem-atico Universita e Politecnico di Torino vol 69 no 1 pp 97ndash1102011

[18] L Toth ldquoOn the number of cyclic subgroups of a finite Abeliangrouprdquo Bulletin Mathematique de la Societe des Sciences Mathe-matiques de Roumanie vol 55 no 103 pp 423ndash428 2012

[19] P J McCarthy Introduction to Arithmetical Functions SpringerNew York NY USA 1986

[20] WGNowak andL Toth ldquoOn the average number of subgroupsof the groupZ

119898timesZ119899rdquo International Journal of NumberTheory

vol 10 pp 363ndash374 2014[21] K Bauer D Sen and P Zvengrowski ldquoA generalized Goursat

lemmardquo httparxivorgabs11090024[22] A Pakapongpun and T Ward ldquoFunctorial orbit countingrdquo

Journal of Integer Sequences vol 12 Article ID 0924 20 pages2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Numbers

group of rank two since 119866 ≃ Z119906times ZV where 119906 = gcd(119898 119899)

and V = lcm(119898 119899) Let 119906 = 1199011198861

1sdot sdot sdot 119901119886119903

119903and V = 119901

1198871

1sdot sdot sdot 119901119887119903

119903be

the prime power factorizations of 119906 and V respectively where0 le 119886119895le 119887119895(1 le 119895 le 119903) Then

119904 (119898 119899) =

119903

prod

119895=1

119904 (119901

119886119895

119895 119901

119887119895

119895) (1)

119904119896(119898 119899) =

119903

prod

119895=1

119904119896119895

(119901

119886119895

119895 119901

119887119895

119895) (2)

where 119896 = 1198961sdot sdot sdot 119896119903and 119896

119895= 119901

119888119895

119895with some exponents 0 le

119888119895le 119886119895+ 119887119895(1 le 119895 le 119903)

Now consider the 119901-group Z119901119886 times Z119901119887 where 0 le 119886 le 119887

This is of rank two for 1 le 119886 le 119887 One has the simple explicitformulae

119904 (119901119886

119901119887

)

=

(119887 minus 119886 + 1) 119901119886+2

minus (119887 minus 119886 minus 1) 119901119886+1

minus (119886 + 119887 + 3) 119901 + (119886 + 119887 + 1)

(119901 minus 1)2

(3)

119904119901119888 (119901119886

119901119887

) =

119901119888+1

minus 1

119901 minus 1

119888 le 119886 le 119887

119901119886+1

minus 1

119901 minus 1

119886 le 119888 le 119887

119901119886+119887minus119888+1

minus 1

119901 minus 1

119886 le 119887 le 119888 le 119886 + 119887

(4)

Formula (3) was derived by Calugareanu [12 Section 4]and recently by Petrillo [13 Proposition 2] using Goursatrsquoslemma for groups Tarnauceanu [14 Proposition 29] and [15Theorem 33] deduced (3) and (4) by a method based onproperties of certain attached matrices

Therefore 119904(119898 119899) and 119904119896(119898 119899) can be computed using (1)

(3) and (2) (4) respectively We deduce other formulas for119904(119898 119899) and 119904

119896(119898 119899) (Theorems 3 and 4) which generalize

(3) and (4) and put them in more compact forms These areconsequences of a simple representation of the subgroups of119866 = Z

119898timesZ119899 given inTheorem 1 This representation might

be known but the only source we could find is the paper[5] where only a special case is treated in a different formMore exactly in [5 Lemma 41] a representation for latticesin Z119899times Z119899of redundancy 2 that is subgroups of Z

119899times Z119899

having index 1198992 is given using matrices in Hermite normalform Theorem 2 gives the invariant factor decompositionsof the subgroups of 119866 We also consider the number ofcyclic subgroups of Z

119898times Z119899(Theorem 5) and the number

of subgroups of a given exponent in Z119899times Z119899(Theorem 8)

Our approach is elementary using only simple group-theoretic and number-theoretic arguments The proofs aregiven in Section 4

Throughout the paper we use the following notationsN = 1 2 N

0= 0 1 2 120591(119899) and 120590(119899) are the

number and the sum respectively of the positive divisors of119899 120595(119899) = 119899prod

119901|119899(1 + 1119901) is the Dedekind function 120596(119899)

stands for the number of distinct prime factors of 119899 120583 is theMobius function 120601 denotes Eulerrsquos totient function and 120577 isthe Riemann zeta function

s

b

a

1

1

0

0

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

Figure 1

2 Subgroups of Z119898times Z119899

The subgroups ofZ119898timesZ119899can be identified and visualized in

the plane with sublattices of the lattice Z119898times Z119899 Every two-

dimensional sublattice is generated by two basis vectors Forexample Figure 1 shows the subgroup ofZ

12timesZ12having the

basis vectors (3 0) and (1 2)This suggests the following representation of the sub-

groups

Theorem 1 For every119898 119899 isin N let

119868119898119899

= (119886 119887 119905) isin N2 times N0 119886 | 119898 119887 | 119899

0 le 119905 le gcd(119886 119899119887

) minus 1

(5)

and for (119886 119887 119905) isin 119868119898119899

define

119867119886119887119905

= (119894119886 +

119895119905119886

gcd (119886 119899119887)

119895119887)

0 le 119894 le

119898

119886

minus 1 0 le 119895 le

119899

119887

minus 1

(6)

Then 119867119886119887119905

is a subgroup of order 119898119899119886119887 of Z119898times Z119899and

the map (119886 119887 119905) 997891rarr 119867119886119887119905

is a bijection between the set 119868119898119899

andthe set of subgroups of Z

119898times Z119899

Note that for the subgroup 119867119886119887119905

the basis vectorsmentioned above are (119886 0) and (119904 119887) where

119904 =

119905119886

gcd (119886 119899119887) (7)

This notation for 119904 will be used also in the rest of thepaper Note also that in the case 119886 = 119898 119887 = 119899 the area of

Journal of Numbers 3

the parallelogram spanned by the basis vectors is 119886119887 exactlythe index of119867

119886119887119905

We say that a subgroup119867 = 119867119886119887119905

is a subproduct ofZ119898times

Z119899if 119867 = 119867

1times 1198672 where 119867

1and 119867

2are subgroups of Z

119898

and Z119899 respectively

Theorem 2 (i) The invariant factor decomposition of thesubgroup119867

119886119887119905is given by

119867119886119887119905

≃ Z120572times Z120573 (8)

where

120572 = (

119898

119886

119899

119887

119899119904

119886119887

) 120573 =

119898119899

a119887120572(9)

satisfying 120572 | 120573(ii) The exponent of the subgroup119867

119886119887119905is 120573

(iii) The subgroup119867119886119887119905

is cyclic if and only if 120572 = 1(iv) The subgroup119867

119886119887119905is a subproduct if and only if 119905 = 0

and 1198671198861198870

= Z119898119886

times Z119899119887

Here 1198671198861198870

is cyclic if and only ifgcd (119898119886 119899119887) = 1

For example for the subgroup represented by Figure 1 onehas 119898 = 119899 = 12 119886 = 3 119887 = 2 119904 = 1 120572 = 2 and 120573 = 12 andthis subgroup is isomorphic to Z

2times Z12 It is not cyclic and

is not a subproductAccording toTheorem 1 the number 119904(119898 119899) of subgroups

of Z119898times Z119899can be obtained by counting the elements of the

set 119868119898119899

We deduce the following

Theorem 3 For every119898 119899 isin N 119904(119898 119899) is given by

119904 (119898 119899) = sum

119886|119898 119887|119899

gcd (119886 119887) (10)

= sum

119889|gcd(119898119899)120601 (119889) 120591 (

119898

119889

) 120591 (

119899

119889

) (11)

= sum

119889|gcd(119898119899)119889120591 (

119898119899

1198892) (12)

Formula (10) is a special case of a formula representingthe number of all subgroups of a class of groups formed ascyclic extensions of cyclic groups deduced by Calhoun [16]and having a laborious proof Note that formula (10) is givenwithout proof in [3 item A054584]

Note also that the function (119898 119899) 997891rarr 119904(119898 119899) is represent-ing a multiplicative arithmetic function of two variables thatis 119904(119898119898

1015840

1198991198991015840

) = 119904(119898 119899)119904(1198981015840

1198991015840

) holds for any 119898 1198991198981015840

1198991015840

isin

N such that gcd(1198981198991198981015840

1198991015840

) = 1 This property which is inconcordance with (1) is a direct consequence of formula (10)See Section 5

Let119873(119886 119887 119888) denote the number of solutions (119909 119910 119911 119905) isinN4 of the system of equations 119909119910 = 119886 119911119905 = 119887 119909119911 = 119888

Theorem 4 For every 119896119898 119899 isin N such that 119896 | 119898119899

119904119896(119898 119899) = sum

119886|119898119887|119899

119898119887119886=119896

gcd (119886 119887)(13)

= sum

119889|gcd(119896119898)119890| gcd (119896119899)119896|119889119890

120601(

119889119890

119896

)

(14)

= sum

119889|gcd(119898119899119896)120601 (119889)119873(

119898

119889

119899

119889

119896

119889

) (15)

The identities (3) and (4) can be easily deduced from eachof the identities given inTheorems 3 and 4 respectively

Theorem 5 Let119898 119899 isin N(i) The number 119888(119898 119899) of cyclic subgroups of Z

119898times Z119899is

given by

119888 (119898 119899) = sum

119886|119898119887|119899

gcd(119898119886119899119887)=1

gcd (119886 119887)(16)

= sum

119886|119898119887|119899

120601 (gcd (119886 119887)) (17)

= sum

119889|gcd(119898119899)(120583 lowast 120601) (119889) 120591 (

119898

119889

) 120591 (

119899

119889

) (18)

= sum

119889|gcd(119898119899)120601 (119889) 120591 (

119898119899

1198892) (19)

(ii)The number of subproducts ofZ119898timesZ119899is 120591(119898)120591(119899) and

the number of its cyclic subproducts is 120591(119898119899)

Formula (17) as a special case of an identity valid forarbitrary finite Abelian groups was derived by the thirdauthor [17 18] using different arguments The function(119898 119899) 997891rarr 119888(119898 119899) is also multiplicative

3 Subgroups of Z119899times Z119899

In the case119898 = 119899 which is of special interest in applicationsthe results given in the previous section can be easily usedWe point out that 119899 997891rarr 119904(119899) = 119904(119899 119899) and 119899 997891rarr 119888(119899) =

119888(119899 119899) are multiplicative arithmetic functions of a singlevariable (sequences [3 items A060724A060648]) They canbe written in the form of Dirichlet convolutions as shown bythe next corollaries

Corollary 6 For every 119899 isin N

119904 (119899) = sum

119889119890=119899

120601 (119889) 1205912

(119890)

= sum

119889119890=119899

119889 120591 (1198902

)

(20)

4 Journal of Numbers

Corollary 7 For every 119899 isin N

119888 (119899) = sum

119889119890=119899

1198892120596(119890)

(21)

= sum

119889119890=119899

120601 (119889) 120591 (1198902

) (22)

Further convolutional representations can also be givenfor example

119904 (119899) = sum

119889119890=119899

120591 (119889) 120595 (119890) 119888 (119899) = sum

119889|119899

120595 (119889) (23)

all of these follow from the Dirichlet-series representationsinfin

sum

119899=1

119904 (119899)

119899119911

=

1205773

(119911) 120577 (119911 minus 1)

120577 (2119911)

(24)

infin

sum

119899=1

119888 (119899)

119899119911

=

1205772

(119911) 120577 (119911 minus 1)

120577 (2119911)

(25)

valid for 119911 isin CR(119911) gt 2Observe that

119904 (119899) = sum

119889|119899

119888 (119889) (119899 isin N) (26)

which is a simple consequence of (23) or of (24) and (25) Italso follows from the next result

Theorem 8 For every 119899 120575 isin N with 120575 | 119899 the number ofsubgroups of exponent 120575 ofZ

119899timesZ119899equals the number of cyclic

subgroups of Z120575times Z120575

4 Proofs

Proof of Theorem 1 Let 119867 be a subgroup of 119866 = Z119898times Z119899

Consider the natural projection 1205872

119866 rarr Z119899given by

1205872(119909 119910) = 119910 Then 120587

2(119867) is a subgroup of Z

119899and there is

a unique divisor 119887 of 119899 such that 1205872(119867) = ⟨119887⟩ = 119895119887 0 le 119895 le

119899119887 minus 1 Let 119904 ge 0 be minimal such that (119904 119887) isin 119867Furthermore consider the natural inclusion 120580

1 Z119898

rarr

119866 given by 1205801(119909) = (119909 0)Then 120580minus1

1(119867) is a subgroup ofZ

119898and

there exists a unique divisor 119886 of119898 such that 120580minus11(119867) = ⟨119886⟩

We show that 119867 = (119894119886 + 119895119904 119895119887) 119894 119895 isin Z Indeed forevery 119894 119895 isin Z (119894119886+119895119904 119895119887) = 119894(119886 0)+119895(119904 119887) isin 119867 On the otherhand for every (119906 V) isin 119867 one has V isin 120587

2(119867) and hence there

is 119895 isin Z such that V = 119895119887 We obtain (119906 minus 119895119904 0) = (119906 V) minus119895(119904 119887) isin 119867 119906minus119895119904 isin 120580

minus1

1(119867) and there is 119894 isin Zwith 119906minus119895119904 = 119894119886

Here a necessary condition is that (119904119899119887 0) isin 119867 (obtainedfor 119894 = 0 and 119895 = 119899119887) that is 119886 | 119904119899b equivalent to119886gcd(119886 119899119887) | 119904 Clearly if this is verified then for the aboverepresentation of 119867 it is enough to take the values 0 le 119894 le

119898119886 minus 1 and 0 le 119895 le 119899119887 minus 1Also dividing 119904 by 119886 we have 119904 = 119886119902 + 119903 with 0 le 119903 lt 119886

and (119903 119887) = (119904 119887) minus 119902(119886 0) isin 119867 showing that 119904 lt 119886 byits minimality Hence 119904 = 119905119886gcd(119886 119899119887) with 0 le 119905 le

gcd(119886 119899119887) minus 1 Thus we obtain the given representationConversely every (119886 119887 119905) isin 119868

119898119899generates a subgroup

119867119886119887119905

of order 119898119899(119886119887) of Z119898

times Z119899and the proof is

complete

Proof of Theorem 2 (i)-(ii) We first determine the exponentof the subgroup 119867

119886119887119905 119867119886119887119905

is generated by (119886 0) and (119904 119887)hence its exponent is the least commonmultiple of the ordersof these two elements The order of (119886 0) is119898119886 To computethe order of (119904 119887) note that119898 | 119903119904 if and only if119898gcd(119898 119904) |

119903Thus the order of (119904 119887) is lcm(119898gcd(119898 119904) 119899119887)We deducethat the exponent of119867

119886119887119905is

lcm(

119898

119886

119898

gcd (119898 119904)

119899

119887

)

= lcm(

119898119899

119899119886

119898119899

119899gcd (119898 119904)

119898119899

119898119887

)

=

119898119899

gcd (119899119886 119899119898 119899119904 119898119887)

=

119898119899

gcd (119898119887 119899119886 119899119904)

= 120573

(27)

For every finite Abelian group the rank of a nontrivialsubgroup is at most the rank of the groupTherefore the rankof 119867119886119887t is 1 or 2 That is 119867

119886119887119905≃ Z119860times Z119861with certain

119860 119861 isin N such that 119860 | 119861 Here the exponent of 119867119886119887119905

equalsthat of Z

119860times Z119861 which is lcm(119860 119861) = 119861 Using (ii) already

proved we deduce that 119861 = 120573 Since the order of 119867119886119887119905

is119860119861 = 119898119899(119886119887) we have 119860 = 119898119899(119886119887120573) = 120572

(iii) According to (i) 119867119886119887119905

≃ Z120572timesZ120573 where 120572 | 120573 Hence

119867119886119887119905

is cyclic if and only if 120572 = 1(iv) The subgroups of Z

119898are of form 119894119886 0 le 119894 le 119898119886 minus

1 where 119886 | 119898 and the properties follow from (6) and (iii)

Proof of Theorem 3 By its definition the number of elementsof the set 119868

119898119899is

sum

119886|119898119887|119899

sum

0le119905legcd(119886119899119887)minus11

= sum

119886|119898119887|119899

gcd(119886 119899119887

) = sum

119886|119898119887|119899

gcd (119886 119887) (28)

representing 119904(119898 119899) This is formula (10)To obtain formula (11) apply the Gauss formula 119899 =

sum119889|119899

120601(119889) (119899 isin N) by writing the following

119904 (119898 119899) = sum

119886|119898119887|119899

sum

119889|gcd(119886119887)120601 (119889) = sum

119886119909=119898

119887119910=119899

sum

119889119894=119886

119889119895=119887

120601 (119889) = sum

119889119894119909=119898

119889119895119910=119899

120601 (119889)

= sum

119889119906=119898

119889V=119899

120601 (119889) sum

119894119909=119906

119895119910=V

1 = sum

119889119906=119898

119889V=119899

120601 (119889) 120591 (119906) 120591 (V)

= sum

119889|gcd(119898119899)120601 (119889) 120591 (

119898

119889

) 120591 (

119899

119889

)

(29)

Journal of Numbers 5

Now (12) follows from (11) by the Busche-Ramanujanidentity (cf [19 Chapter 1])

120591 (119898) 120591 (119899) = sum

119889|gcd(119898119899)120591 (

119898119899

1198892) (119898 119899 isin N) (30)

Proof of Theorem 4 According toTheorem 1

119904119896(119898 119899) = sum

119886|119898119887|119899

119898119899119886119887=119896

gcd(119886 119899119887

) (31)

giving (13) which can be written by Gaussrsquo formula again as

119904119896(119898 119899) = sum

119886|119898119887|119899

119898119887119886=119896

sum

119888|119886119888|119887

120601 (119888) = sum

119888119894119909=119898

119888119895119910=119899

119888119895119909=119896

120601 (119888)

= sum

119889119894=119898

119890119910=119899

sum

119888119909=119889

119888119895=119890

119888119895119909=119896

120601 (119888)

(32)

where in the inner sum one has 119888 = 119889119890119896 and obtains (14)Now to get (15) write (32) as

119904119896(119898 119899) = sum

119888119906=119898

119888V=119899119888119908=119896

120601 (119888) sum

119894119909=119906

119895119910=V119895119909=119908

1 = sum

119888119906=119898

119888V=119899119888119908=119896

120601 (119888)119873 (119906 V 119908)

= sum

119888|gcd(119898119899119896)120601 (119888)119873(

119898

119888

119899

119888

119896

119888

)

(33)

and the proof is complete

Proof of Theorem 5 (i) According to Theorems 1 and 2(iii)and using that sum

119889|119899120583(119889) = 1 or 0 and according to 119899 = 1 or

119899 gt 1

119888 (119898 119899) = sum

119886|119898119887|119899

sum

1le119904le119886

119886119887|119899119904

gcd(119898119886119899119887119899119904119886119887)=1

1 = sum

119886119909=119898

119887119910=119899

sum

1le119904le119886

119886119903=119910119904

gcd(119909119910119903)=1

1

= sum

119886119909=119898

119887119910=119899

sum

1le119904le119886

119886119903=119910119904

sum

119890|gcd(119909119910119903)

120583 (119890) = sum

119886119890119894=119898

119887119890119895=119899

120583 (119890) sum

1le119904le119886

119886gcd(119886119895)|119904

1

(34)

where the inner sum is gcd(119886 119895) Hence

119888 (119898 119899) = sum

119886119890119894=119898

119887119890119895=119899

120583 (119890) gcd (119886 119895) (35)

Now regrouping the terms according to 119890119894 = 119911 and 119887119890 = 119905

we obtain

119888 (119898 119899) = sum

119886119911=119898

119895119905=119899

gcd (119886 119895) sum119890119894=119911

119887119890=119905

120583 (119890) = sum

119886119911=119898

119895119905=119899

gcd (119886 119895) sum

119890|gcd(119911119905)120583 (119890)

= sum

119886119911=119898

119895119905=119899

gcd(119911119905)=1

gcd (119886 119895)

(36)

which is (16)

The next results follow applying Gaussrsquo formula andthe Busche-Ramanujan formula similar to the proof ofTheorem 3

(ii) For the subproducts 1198671198861198870

the values 119886 | 119898 and 119887 | 119899

can be chosen arbitrary and it follows at once that the numberof subproducts is 120591(119898)120591(119899)Thenumber of cyclic subproductsis

sum

119886|119898

119887|119899

gcd(119898119886119899119887)=1

1 = sum

119886119909=119898

119887119910=119899

gcd(119909119910)=1

1 = sum

119886119909=119898

119887119910=119899

sum

119890|gcd(119909119910)

120583 (119890)

= sum

119890119860=119898

119890119861=119899

120583 (119890) 120591 (119860) 120591 (119861)

= sum

119890|gcd(119898119899)120583 (119890) 120591 (

119898

119890

) 120591 (

119899

119890

) = 120591 (119898119899)

(37)

by the inverse Busche-Ramanujan identity

Proof of Theorem 8 According to Theorem 2(ii) the num-ber of subgroups of exponent 120575 of Z

119899times Z119899is

119864120575(119899) = sum

119886|119899119887|119899

sum

1le119904le119886

119886119887|119899119904

119899gcd(119886119887119904)=120575

1 = sum

119886119909=119899

119887119910=119899

sum

1le119904le119886

119886119903=119910119904

gcd(119886119887119904)=119899120575

1

(38)

Write 119886 = 1198861119899120575 119887 = 119887

1119899120575 and 119904 = 119904

1119899120575 with

gcd(1198861 1198871 1199041) = 1 We deduce similar to the proof of

Theorem 5(i) that

119864120575(119899) = sum

119890119894119909=120575

119890119895119910=120575

120583 (119890) gcd (119894 119910) (39)

which is exactly 119888(120575 120575) = 119888(120575) (cf (35))

5 Further Remarks

(1) As mentioned in Section 2 the functions (119898 119899) 997891rarr

119904(119898 119899) and (119898 119899) 997891rarr 119888(119898 119899) are multiplicative func-tions of two variablesThis follows easily from formu-lae (10) and (17) respectively Namely according tothose formulae 119904(119898 119899) and 119888(119898 119899) are two variablesDirichlet convolutions of the functions (119898 119899) 997891rarr

gcd(119898 119899) and (119898 119899) 997891rarr 120601(gcd(119898 119899)) respectivelywith the constant 1 function all multiplicative Sinceconvolution preserves the multiplicativity we deducethat 119904(119898 119899) and 119888(119898 119899) are alsomultiplicative See [17Section 2] for details

(2) Asymptotic formulas with sharp error terms for thesums sum

119898119899le119909119904(119898 119899) and sum

119898119899le119909119888(119898 119899) were given in

the paper [20](3) For any finite groups 119860 and 119861 a subgroup 119862 of 119860 times 119861

is cyclic if and only if 120580minus11(119862) and 120580

minus1

2(119862) have coprime

orders where 1205801and 1205802are the natural inclusions ([21

Theorem 42]) In the case 119860 = Z119898 119861 = Z

119899 and

119862 = 119867119886119887119905

one has 120580minus11(119862) = 119898119886 and 120580minus1

2(119862) =

gcd(119899119887 119899119904119886119887) and the characterization of the cyclicsubgroups 119867

119886119887119905given in Theorem 2(iii) can be

6 Journal of Numbers

obtained also in this way It turns out that regardingthe sublattice 119867

119886119887119905is cyclic if and only if the

numbers of points on the horizontal and vertical axesrespectively are relatively prime Note that in the case119898 = 119899 the above condition reads 119899gcd(119886 119887 119904) = 119886119887Thus it is necessary that 119899 | 119886119887 The subgroup onFigure 1 is not cyclic

(4) Note also the next formula for the number of cyclicsubgroups of Z

119899times Z119899 derived in [22 Example 2]

119888 (119899) = sum

lcm(119889119890)=119899gcd (119889 119890) (119899 isin N) (40)

where the sum is over all ordered pairs (119889 119890) such thatlcm(119889 119890) = 119899 For a short direct proof of (40) write119889 = ℓ119886 119890 = ℓ119887 with gcd(119886 119887) = 1 Then gcd(119889 119890) = ℓlcm(119889 119890) = ℓ119886119887 and obtain

sum

lcm(119889119890)=119899gcd (119889 119890)

= sum

ℓ119886119887=119899

gcd(119886119887)=1

ℓ = sum

ℓ119896=119899

ℓ sum

119886119887=119896

gcd(119886119887)=1

1 = sum

ℓ119896=119899

ℓ2120596(119896)

= 119888 (119899)

(41)

according to (21)(5) Every subgroup119870 ofZtimesZ has the representation119870 =

(119894119886 + 119895119904 119895119887) 119894 119895 isin Z where 0 le 119904 le 119886 and 0 le 119887

are unique integers This follows like in the proof ofTheorem 1 Furthermore in the case 119886 119887 ge 1 0 le 119904 le

119886minus 1 the index of119870 is 119886119887 and one obtains at once thatthe number of subgroups 119870 having index 119899 (119899 isin N)is sum119886119887=119899

sum0le119904le119886minus1

1 = sum119886119887=119899

119886 = 120590(119899) mentioned inSection 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Nicki Holighaus was partially supported by the AustrianScience Fund (FWF) START-Project FLAME (Y551-N13)Laszlo Toth gratefully acknowledges support from the Aus-trian Science Fund (FWF) under Project no M1376-N18Christoph Wiesmeyr was partially supported by EU FETOpen Grant UNLocX (255931)

References

[1] M J Grady ldquoA group theoretic approach to a famous partitionformulardquo The American Mathematical Monthly vol 112 no 7pp 645ndash651 2005

[2] Y M Zou ldquoGaussian binomials and the number of sublatticesrdquoActa Crystallographica vol 62 no 5 pp 409ndash410 2006

[3] ldquoThe On-Line Encyclopedia of Integer Sequencesrdquo httpoeisorg

[4] K Grochenig Foundations of Time-Frequency Analysis Appliedand Numerical Harmonic Analysis Birkhauser Boston MassUSA 2001

[5] G Kutyniok and T Strohmer ldquoWilson bases for general time-frequency latticesrdquo SIAM Journal onMathematical Analysis vol37 no 3 pp 685ndash711 (electronic) 2005

[6] A J van Leest Non-separable Gabor schemes their design andimplementation [PhD thesis] Technische Universiteit Eind-hoven Eindhoven The Netherlands 2001

[7] T Strohmer ldquoNumerical algorithms for discrete Gabor expan-sionsrdquo in Gabor Analysis and Algorithms Theory and Applica-tions H G Feichtinger and T Strohmer Eds pp 267ndash294Birkhauser Boston Mass USA 1998

[8] A Machı Groups An Introduction to Ideas and Methods of theTheory of Groups Springer 2012

[9] J J Rotman An Introduction to theTheory of Groups vol 148 ofGraduate Texts in Mathematics Springer New York NY USA4th edition 1995

[10] R Schmidt Subgroup Lattices of Groups deGruyter Expositionsin Mathematics 14 de Gruyter Berlin Germany 1994

[11] M Suzuki ldquoOn the lattice of subgroups of finite groupsrdquoTransactions of the American Mathematical Society vol 70 pp345ndash371 1951

[12] G Calugareanu ldquoThe total number of subgroups of a finiteabelian grouprdquo Scientiae Mathematicae Japonicae vol 60 no1 pp 157ndash167 2004

[13] J Petrillo ldquoCounting subgroups in a direct product of finitecyclic groupsrdquo College Mathematics Journal vol 42 no 3 pp215ndash222 2011

[14] M Tarnauceanu ldquoA new method of proving some classicaltheorems of abelian groupsrdquo Southeast Asian Bulletin of Mathe-matics vol 31 no 6 pp 1191ndash1203 2007

[15] M Tarnauceanu ldquoAn arithmetic method of counting the sub-groups of a finite abelian grouprdquo Bulletin Mathematiques de laSociete des SciencesMathematiques de Roumanie vol 53 no 101pp 373ndash386 2010

[16] W C Calhoun ldquoCounting the subgroups of some finite groupsrdquoThe American Mathematical Monthly vol 94 no 1 pp 54ndash591987

[17] L Toth ldquoMenons identity and arithmetical sums representingfunctions of several variablesrdquoRendiconti del SeminarioMatem-atico Universita e Politecnico di Torino vol 69 no 1 pp 97ndash1102011

[18] L Toth ldquoOn the number of cyclic subgroups of a finite Abeliangrouprdquo Bulletin Mathematique de la Societe des Sciences Mathe-matiques de Roumanie vol 55 no 103 pp 423ndash428 2012

[19] P J McCarthy Introduction to Arithmetical Functions SpringerNew York NY USA 1986

[20] WGNowak andL Toth ldquoOn the average number of subgroupsof the groupZ

119898timesZ119899rdquo International Journal of NumberTheory

vol 10 pp 363ndash374 2014[21] K Bauer D Sen and P Zvengrowski ldquoA generalized Goursat

lemmardquo httparxivorgabs11090024[22] A Pakapongpun and T Ward ldquoFunctorial orbit countingrdquo

Journal of Integer Sequences vol 12 Article ID 0924 20 pages2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Numbers 3

the parallelogram spanned by the basis vectors is 119886119887 exactlythe index of119867

119886119887119905

We say that a subgroup119867 = 119867119886119887119905

is a subproduct ofZ119898times

Z119899if 119867 = 119867

1times 1198672 where 119867

1and 119867

2are subgroups of Z

119898

and Z119899 respectively

Theorem 2 (i) The invariant factor decomposition of thesubgroup119867

119886119887119905is given by

119867119886119887119905

≃ Z120572times Z120573 (8)

where

120572 = (

119898

119886

119899

119887

119899119904

119886119887

) 120573 =

119898119899

a119887120572(9)

satisfying 120572 | 120573(ii) The exponent of the subgroup119867

119886119887119905is 120573

(iii) The subgroup119867119886119887119905

is cyclic if and only if 120572 = 1(iv) The subgroup119867

119886119887119905is a subproduct if and only if 119905 = 0

and 1198671198861198870

= Z119898119886

times Z119899119887

Here 1198671198861198870

is cyclic if and only ifgcd (119898119886 119899119887) = 1

For example for the subgroup represented by Figure 1 onehas 119898 = 119899 = 12 119886 = 3 119887 = 2 119904 = 1 120572 = 2 and 120573 = 12 andthis subgroup is isomorphic to Z

2times Z12 It is not cyclic and

is not a subproductAccording toTheorem 1 the number 119904(119898 119899) of subgroups

of Z119898times Z119899can be obtained by counting the elements of the

set 119868119898119899

We deduce the following

Theorem 3 For every119898 119899 isin N 119904(119898 119899) is given by

119904 (119898 119899) = sum

119886|119898 119887|119899

gcd (119886 119887) (10)

= sum

119889|gcd(119898119899)120601 (119889) 120591 (

119898

119889

) 120591 (

119899

119889

) (11)

= sum

119889|gcd(119898119899)119889120591 (

119898119899

1198892) (12)

Formula (10) is a special case of a formula representingthe number of all subgroups of a class of groups formed ascyclic extensions of cyclic groups deduced by Calhoun [16]and having a laborious proof Note that formula (10) is givenwithout proof in [3 item A054584]

Note also that the function (119898 119899) 997891rarr 119904(119898 119899) is represent-ing a multiplicative arithmetic function of two variables thatis 119904(119898119898

1015840

1198991198991015840

) = 119904(119898 119899)119904(1198981015840

1198991015840

) holds for any 119898 1198991198981015840

1198991015840

isin

N such that gcd(1198981198991198981015840

1198991015840

) = 1 This property which is inconcordance with (1) is a direct consequence of formula (10)See Section 5

Let119873(119886 119887 119888) denote the number of solutions (119909 119910 119911 119905) isinN4 of the system of equations 119909119910 = 119886 119911119905 = 119887 119909119911 = 119888

Theorem 4 For every 119896119898 119899 isin N such that 119896 | 119898119899

119904119896(119898 119899) = sum

119886|119898119887|119899

119898119887119886=119896

gcd (119886 119887)(13)

= sum

119889|gcd(119896119898)119890| gcd (119896119899)119896|119889119890

120601(

119889119890

119896

)

(14)

= sum

119889|gcd(119898119899119896)120601 (119889)119873(

119898

119889

119899

119889

119896

119889

) (15)

The identities (3) and (4) can be easily deduced from eachof the identities given inTheorems 3 and 4 respectively

Theorem 5 Let119898 119899 isin N(i) The number 119888(119898 119899) of cyclic subgroups of Z

119898times Z119899is

given by

119888 (119898 119899) = sum

119886|119898119887|119899

gcd(119898119886119899119887)=1

gcd (119886 119887)(16)

= sum

119886|119898119887|119899

120601 (gcd (119886 119887)) (17)

= sum

119889|gcd(119898119899)(120583 lowast 120601) (119889) 120591 (

119898

119889

) 120591 (

119899

119889

) (18)

= sum

119889|gcd(119898119899)120601 (119889) 120591 (

119898119899

1198892) (19)

(ii)The number of subproducts ofZ119898timesZ119899is 120591(119898)120591(119899) and

the number of its cyclic subproducts is 120591(119898119899)

Formula (17) as a special case of an identity valid forarbitrary finite Abelian groups was derived by the thirdauthor [17 18] using different arguments The function(119898 119899) 997891rarr 119888(119898 119899) is also multiplicative

3 Subgroups of Z119899times Z119899

In the case119898 = 119899 which is of special interest in applicationsthe results given in the previous section can be easily usedWe point out that 119899 997891rarr 119904(119899) = 119904(119899 119899) and 119899 997891rarr 119888(119899) =

119888(119899 119899) are multiplicative arithmetic functions of a singlevariable (sequences [3 items A060724A060648]) They canbe written in the form of Dirichlet convolutions as shown bythe next corollaries

Corollary 6 For every 119899 isin N

119904 (119899) = sum

119889119890=119899

120601 (119889) 1205912

(119890)

= sum

119889119890=119899

119889 120591 (1198902

)

(20)

4 Journal of Numbers

Corollary 7 For every 119899 isin N

119888 (119899) = sum

119889119890=119899

1198892120596(119890)

(21)

= sum

119889119890=119899

120601 (119889) 120591 (1198902

) (22)

Further convolutional representations can also be givenfor example

119904 (119899) = sum

119889119890=119899

120591 (119889) 120595 (119890) 119888 (119899) = sum

119889|119899

120595 (119889) (23)

all of these follow from the Dirichlet-series representationsinfin

sum

119899=1

119904 (119899)

119899119911

=

1205773

(119911) 120577 (119911 minus 1)

120577 (2119911)

(24)

infin

sum

119899=1

119888 (119899)

119899119911

=

1205772

(119911) 120577 (119911 minus 1)

120577 (2119911)

(25)

valid for 119911 isin CR(119911) gt 2Observe that

119904 (119899) = sum

119889|119899

119888 (119889) (119899 isin N) (26)

which is a simple consequence of (23) or of (24) and (25) Italso follows from the next result

Theorem 8 For every 119899 120575 isin N with 120575 | 119899 the number ofsubgroups of exponent 120575 ofZ

119899timesZ119899equals the number of cyclic

subgroups of Z120575times Z120575

4 Proofs

Proof of Theorem 1 Let 119867 be a subgroup of 119866 = Z119898times Z119899

Consider the natural projection 1205872

119866 rarr Z119899given by

1205872(119909 119910) = 119910 Then 120587

2(119867) is a subgroup of Z

119899and there is

a unique divisor 119887 of 119899 such that 1205872(119867) = ⟨119887⟩ = 119895119887 0 le 119895 le

119899119887 minus 1 Let 119904 ge 0 be minimal such that (119904 119887) isin 119867Furthermore consider the natural inclusion 120580

1 Z119898

rarr

119866 given by 1205801(119909) = (119909 0)Then 120580minus1

1(119867) is a subgroup ofZ

119898and

there exists a unique divisor 119886 of119898 such that 120580minus11(119867) = ⟨119886⟩

We show that 119867 = (119894119886 + 119895119904 119895119887) 119894 119895 isin Z Indeed forevery 119894 119895 isin Z (119894119886+119895119904 119895119887) = 119894(119886 0)+119895(119904 119887) isin 119867 On the otherhand for every (119906 V) isin 119867 one has V isin 120587

2(119867) and hence there

is 119895 isin Z such that V = 119895119887 We obtain (119906 minus 119895119904 0) = (119906 V) minus119895(119904 119887) isin 119867 119906minus119895119904 isin 120580

minus1

1(119867) and there is 119894 isin Zwith 119906minus119895119904 = 119894119886

Here a necessary condition is that (119904119899119887 0) isin 119867 (obtainedfor 119894 = 0 and 119895 = 119899119887) that is 119886 | 119904119899b equivalent to119886gcd(119886 119899119887) | 119904 Clearly if this is verified then for the aboverepresentation of 119867 it is enough to take the values 0 le 119894 le

119898119886 minus 1 and 0 le 119895 le 119899119887 minus 1Also dividing 119904 by 119886 we have 119904 = 119886119902 + 119903 with 0 le 119903 lt 119886

and (119903 119887) = (119904 119887) minus 119902(119886 0) isin 119867 showing that 119904 lt 119886 byits minimality Hence 119904 = 119905119886gcd(119886 119899119887) with 0 le 119905 le

gcd(119886 119899119887) minus 1 Thus we obtain the given representationConversely every (119886 119887 119905) isin 119868

119898119899generates a subgroup

119867119886119887119905

of order 119898119899(119886119887) of Z119898

times Z119899and the proof is

complete

Proof of Theorem 2 (i)-(ii) We first determine the exponentof the subgroup 119867

119886119887119905 119867119886119887119905

is generated by (119886 0) and (119904 119887)hence its exponent is the least commonmultiple of the ordersof these two elements The order of (119886 0) is119898119886 To computethe order of (119904 119887) note that119898 | 119903119904 if and only if119898gcd(119898 119904) |

119903Thus the order of (119904 119887) is lcm(119898gcd(119898 119904) 119899119887)We deducethat the exponent of119867

119886119887119905is

lcm(

119898

119886

119898

gcd (119898 119904)

119899

119887

)

= lcm(

119898119899

119899119886

119898119899

119899gcd (119898 119904)

119898119899

119898119887

)

=

119898119899

gcd (119899119886 119899119898 119899119904 119898119887)

=

119898119899

gcd (119898119887 119899119886 119899119904)

= 120573

(27)

For every finite Abelian group the rank of a nontrivialsubgroup is at most the rank of the groupTherefore the rankof 119867119886119887t is 1 or 2 That is 119867

119886119887119905≃ Z119860times Z119861with certain

119860 119861 isin N such that 119860 | 119861 Here the exponent of 119867119886119887119905

equalsthat of Z

119860times Z119861 which is lcm(119860 119861) = 119861 Using (ii) already

proved we deduce that 119861 = 120573 Since the order of 119867119886119887119905

is119860119861 = 119898119899(119886119887) we have 119860 = 119898119899(119886119887120573) = 120572

(iii) According to (i) 119867119886119887119905

≃ Z120572timesZ120573 where 120572 | 120573 Hence

119867119886119887119905

is cyclic if and only if 120572 = 1(iv) The subgroups of Z

119898are of form 119894119886 0 le 119894 le 119898119886 minus

1 where 119886 | 119898 and the properties follow from (6) and (iii)

Proof of Theorem 3 By its definition the number of elementsof the set 119868

119898119899is

sum

119886|119898119887|119899

sum

0le119905legcd(119886119899119887)minus11

= sum

119886|119898119887|119899

gcd(119886 119899119887

) = sum

119886|119898119887|119899

gcd (119886 119887) (28)

representing 119904(119898 119899) This is formula (10)To obtain formula (11) apply the Gauss formula 119899 =

sum119889|119899

120601(119889) (119899 isin N) by writing the following

119904 (119898 119899) = sum

119886|119898119887|119899

sum

119889|gcd(119886119887)120601 (119889) = sum

119886119909=119898

119887119910=119899

sum

119889119894=119886

119889119895=119887

120601 (119889) = sum

119889119894119909=119898

119889119895119910=119899

120601 (119889)

= sum

119889119906=119898

119889V=119899

120601 (119889) sum

119894119909=119906

119895119910=V

1 = sum

119889119906=119898

119889V=119899

120601 (119889) 120591 (119906) 120591 (V)

= sum

119889|gcd(119898119899)120601 (119889) 120591 (

119898

119889

) 120591 (

119899

119889

)

(29)

Journal of Numbers 5

Now (12) follows from (11) by the Busche-Ramanujanidentity (cf [19 Chapter 1])

120591 (119898) 120591 (119899) = sum

119889|gcd(119898119899)120591 (

119898119899

1198892) (119898 119899 isin N) (30)

Proof of Theorem 4 According toTheorem 1

119904119896(119898 119899) = sum

119886|119898119887|119899

119898119899119886119887=119896

gcd(119886 119899119887

) (31)

giving (13) which can be written by Gaussrsquo formula again as

119904119896(119898 119899) = sum

119886|119898119887|119899

119898119887119886=119896

sum

119888|119886119888|119887

120601 (119888) = sum

119888119894119909=119898

119888119895119910=119899

119888119895119909=119896

120601 (119888)

= sum

119889119894=119898

119890119910=119899

sum

119888119909=119889

119888119895=119890

119888119895119909=119896

120601 (119888)

(32)

where in the inner sum one has 119888 = 119889119890119896 and obtains (14)Now to get (15) write (32) as

119904119896(119898 119899) = sum

119888119906=119898

119888V=119899119888119908=119896

120601 (119888) sum

119894119909=119906

119895119910=V119895119909=119908

1 = sum

119888119906=119898

119888V=119899119888119908=119896

120601 (119888)119873 (119906 V 119908)

= sum

119888|gcd(119898119899119896)120601 (119888)119873(

119898

119888

119899

119888

119896

119888

)

(33)

and the proof is complete

Proof of Theorem 5 (i) According to Theorems 1 and 2(iii)and using that sum

119889|119899120583(119889) = 1 or 0 and according to 119899 = 1 or

119899 gt 1

119888 (119898 119899) = sum

119886|119898119887|119899

sum

1le119904le119886

119886119887|119899119904

gcd(119898119886119899119887119899119904119886119887)=1

1 = sum

119886119909=119898

119887119910=119899

sum

1le119904le119886

119886119903=119910119904

gcd(119909119910119903)=1

1

= sum

119886119909=119898

119887119910=119899

sum

1le119904le119886

119886119903=119910119904

sum

119890|gcd(119909119910119903)

120583 (119890) = sum

119886119890119894=119898

119887119890119895=119899

120583 (119890) sum

1le119904le119886

119886gcd(119886119895)|119904

1

(34)

where the inner sum is gcd(119886 119895) Hence

119888 (119898 119899) = sum

119886119890119894=119898

119887119890119895=119899

120583 (119890) gcd (119886 119895) (35)

Now regrouping the terms according to 119890119894 = 119911 and 119887119890 = 119905

we obtain

119888 (119898 119899) = sum

119886119911=119898

119895119905=119899

gcd (119886 119895) sum119890119894=119911

119887119890=119905

120583 (119890) = sum

119886119911=119898

119895119905=119899

gcd (119886 119895) sum

119890|gcd(119911119905)120583 (119890)

= sum

119886119911=119898

119895119905=119899

gcd(119911119905)=1

gcd (119886 119895)

(36)

which is (16)

The next results follow applying Gaussrsquo formula andthe Busche-Ramanujan formula similar to the proof ofTheorem 3

(ii) For the subproducts 1198671198861198870

the values 119886 | 119898 and 119887 | 119899

can be chosen arbitrary and it follows at once that the numberof subproducts is 120591(119898)120591(119899)Thenumber of cyclic subproductsis

sum

119886|119898

119887|119899

gcd(119898119886119899119887)=1

1 = sum

119886119909=119898

119887119910=119899

gcd(119909119910)=1

1 = sum

119886119909=119898

119887119910=119899

sum

119890|gcd(119909119910)

120583 (119890)

= sum

119890119860=119898

119890119861=119899

120583 (119890) 120591 (119860) 120591 (119861)

= sum

119890|gcd(119898119899)120583 (119890) 120591 (

119898

119890

) 120591 (

119899

119890

) = 120591 (119898119899)

(37)

by the inverse Busche-Ramanujan identity

Proof of Theorem 8 According to Theorem 2(ii) the num-ber of subgroups of exponent 120575 of Z

119899times Z119899is

119864120575(119899) = sum

119886|119899119887|119899

sum

1le119904le119886

119886119887|119899119904

119899gcd(119886119887119904)=120575

1 = sum

119886119909=119899

119887119910=119899

sum

1le119904le119886

119886119903=119910119904

gcd(119886119887119904)=119899120575

1

(38)

Write 119886 = 1198861119899120575 119887 = 119887

1119899120575 and 119904 = 119904

1119899120575 with

gcd(1198861 1198871 1199041) = 1 We deduce similar to the proof of

Theorem 5(i) that

119864120575(119899) = sum

119890119894119909=120575

119890119895119910=120575

120583 (119890) gcd (119894 119910) (39)

which is exactly 119888(120575 120575) = 119888(120575) (cf (35))

5 Further Remarks

(1) As mentioned in Section 2 the functions (119898 119899) 997891rarr

119904(119898 119899) and (119898 119899) 997891rarr 119888(119898 119899) are multiplicative func-tions of two variablesThis follows easily from formu-lae (10) and (17) respectively Namely according tothose formulae 119904(119898 119899) and 119888(119898 119899) are two variablesDirichlet convolutions of the functions (119898 119899) 997891rarr

gcd(119898 119899) and (119898 119899) 997891rarr 120601(gcd(119898 119899)) respectivelywith the constant 1 function all multiplicative Sinceconvolution preserves the multiplicativity we deducethat 119904(119898 119899) and 119888(119898 119899) are alsomultiplicative See [17Section 2] for details

(2) Asymptotic formulas with sharp error terms for thesums sum

119898119899le119909119904(119898 119899) and sum

119898119899le119909119888(119898 119899) were given in

the paper [20](3) For any finite groups 119860 and 119861 a subgroup 119862 of 119860 times 119861

is cyclic if and only if 120580minus11(119862) and 120580

minus1

2(119862) have coprime

orders where 1205801and 1205802are the natural inclusions ([21

Theorem 42]) In the case 119860 = Z119898 119861 = Z

119899 and

119862 = 119867119886119887119905

one has 120580minus11(119862) = 119898119886 and 120580minus1

2(119862) =

gcd(119899119887 119899119904119886119887) and the characterization of the cyclicsubgroups 119867

119886119887119905given in Theorem 2(iii) can be

6 Journal of Numbers

obtained also in this way It turns out that regardingthe sublattice 119867

119886119887119905is cyclic if and only if the

numbers of points on the horizontal and vertical axesrespectively are relatively prime Note that in the case119898 = 119899 the above condition reads 119899gcd(119886 119887 119904) = 119886119887Thus it is necessary that 119899 | 119886119887 The subgroup onFigure 1 is not cyclic

(4) Note also the next formula for the number of cyclicsubgroups of Z

119899times Z119899 derived in [22 Example 2]

119888 (119899) = sum

lcm(119889119890)=119899gcd (119889 119890) (119899 isin N) (40)

where the sum is over all ordered pairs (119889 119890) such thatlcm(119889 119890) = 119899 For a short direct proof of (40) write119889 = ℓ119886 119890 = ℓ119887 with gcd(119886 119887) = 1 Then gcd(119889 119890) = ℓlcm(119889 119890) = ℓ119886119887 and obtain

sum

lcm(119889119890)=119899gcd (119889 119890)

= sum

ℓ119886119887=119899

gcd(119886119887)=1

ℓ = sum

ℓ119896=119899

ℓ sum

119886119887=119896

gcd(119886119887)=1

1 = sum

ℓ119896=119899

ℓ2120596(119896)

= 119888 (119899)

(41)

according to (21)(5) Every subgroup119870 ofZtimesZ has the representation119870 =

(119894119886 + 119895119904 119895119887) 119894 119895 isin Z where 0 le 119904 le 119886 and 0 le 119887

are unique integers This follows like in the proof ofTheorem 1 Furthermore in the case 119886 119887 ge 1 0 le 119904 le

119886minus 1 the index of119870 is 119886119887 and one obtains at once thatthe number of subgroups 119870 having index 119899 (119899 isin N)is sum119886119887=119899

sum0le119904le119886minus1

1 = sum119886119887=119899

119886 = 120590(119899) mentioned inSection 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Nicki Holighaus was partially supported by the AustrianScience Fund (FWF) START-Project FLAME (Y551-N13)Laszlo Toth gratefully acknowledges support from the Aus-trian Science Fund (FWF) under Project no M1376-N18Christoph Wiesmeyr was partially supported by EU FETOpen Grant UNLocX (255931)

References

[1] M J Grady ldquoA group theoretic approach to a famous partitionformulardquo The American Mathematical Monthly vol 112 no 7pp 645ndash651 2005

[2] Y M Zou ldquoGaussian binomials and the number of sublatticesrdquoActa Crystallographica vol 62 no 5 pp 409ndash410 2006

[3] ldquoThe On-Line Encyclopedia of Integer Sequencesrdquo httpoeisorg

[4] K Grochenig Foundations of Time-Frequency Analysis Appliedand Numerical Harmonic Analysis Birkhauser Boston MassUSA 2001

[5] G Kutyniok and T Strohmer ldquoWilson bases for general time-frequency latticesrdquo SIAM Journal onMathematical Analysis vol37 no 3 pp 685ndash711 (electronic) 2005

[6] A J van Leest Non-separable Gabor schemes their design andimplementation [PhD thesis] Technische Universiteit Eind-hoven Eindhoven The Netherlands 2001

[7] T Strohmer ldquoNumerical algorithms for discrete Gabor expan-sionsrdquo in Gabor Analysis and Algorithms Theory and Applica-tions H G Feichtinger and T Strohmer Eds pp 267ndash294Birkhauser Boston Mass USA 1998

[8] A Machı Groups An Introduction to Ideas and Methods of theTheory of Groups Springer 2012

[9] J J Rotman An Introduction to theTheory of Groups vol 148 ofGraduate Texts in Mathematics Springer New York NY USA4th edition 1995

[10] R Schmidt Subgroup Lattices of Groups deGruyter Expositionsin Mathematics 14 de Gruyter Berlin Germany 1994

[11] M Suzuki ldquoOn the lattice of subgroups of finite groupsrdquoTransactions of the American Mathematical Society vol 70 pp345ndash371 1951

[12] G Calugareanu ldquoThe total number of subgroups of a finiteabelian grouprdquo Scientiae Mathematicae Japonicae vol 60 no1 pp 157ndash167 2004

[13] J Petrillo ldquoCounting subgroups in a direct product of finitecyclic groupsrdquo College Mathematics Journal vol 42 no 3 pp215ndash222 2011

[14] M Tarnauceanu ldquoA new method of proving some classicaltheorems of abelian groupsrdquo Southeast Asian Bulletin of Mathe-matics vol 31 no 6 pp 1191ndash1203 2007

[15] M Tarnauceanu ldquoAn arithmetic method of counting the sub-groups of a finite abelian grouprdquo Bulletin Mathematiques de laSociete des SciencesMathematiques de Roumanie vol 53 no 101pp 373ndash386 2010

[16] W C Calhoun ldquoCounting the subgroups of some finite groupsrdquoThe American Mathematical Monthly vol 94 no 1 pp 54ndash591987

[17] L Toth ldquoMenons identity and arithmetical sums representingfunctions of several variablesrdquoRendiconti del SeminarioMatem-atico Universita e Politecnico di Torino vol 69 no 1 pp 97ndash1102011

[18] L Toth ldquoOn the number of cyclic subgroups of a finite Abeliangrouprdquo Bulletin Mathematique de la Societe des Sciences Mathe-matiques de Roumanie vol 55 no 103 pp 423ndash428 2012

[19] P J McCarthy Introduction to Arithmetical Functions SpringerNew York NY USA 1986

[20] WGNowak andL Toth ldquoOn the average number of subgroupsof the groupZ

119898timesZ119899rdquo International Journal of NumberTheory

vol 10 pp 363ndash374 2014[21] K Bauer D Sen and P Zvengrowski ldquoA generalized Goursat

lemmardquo httparxivorgabs11090024[22] A Pakapongpun and T Ward ldquoFunctorial orbit countingrdquo

Journal of Integer Sequences vol 12 Article ID 0924 20 pages2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Numbers

Corollary 7 For every 119899 isin N

119888 (119899) = sum

119889119890=119899

1198892120596(119890)

(21)

= sum

119889119890=119899

120601 (119889) 120591 (1198902

) (22)

Further convolutional representations can also be givenfor example

119904 (119899) = sum

119889119890=119899

120591 (119889) 120595 (119890) 119888 (119899) = sum

119889|119899

120595 (119889) (23)

all of these follow from the Dirichlet-series representationsinfin

sum

119899=1

119904 (119899)

119899119911

=

1205773

(119911) 120577 (119911 minus 1)

120577 (2119911)

(24)

infin

sum

119899=1

119888 (119899)

119899119911

=

1205772

(119911) 120577 (119911 minus 1)

120577 (2119911)

(25)

valid for 119911 isin CR(119911) gt 2Observe that

119904 (119899) = sum

119889|119899

119888 (119889) (119899 isin N) (26)

which is a simple consequence of (23) or of (24) and (25) Italso follows from the next result

Theorem 8 For every 119899 120575 isin N with 120575 | 119899 the number ofsubgroups of exponent 120575 ofZ

119899timesZ119899equals the number of cyclic

subgroups of Z120575times Z120575

4 Proofs

Proof of Theorem 1 Let 119867 be a subgroup of 119866 = Z119898times Z119899

Consider the natural projection 1205872

119866 rarr Z119899given by

1205872(119909 119910) = 119910 Then 120587

2(119867) is a subgroup of Z

119899and there is

a unique divisor 119887 of 119899 such that 1205872(119867) = ⟨119887⟩ = 119895119887 0 le 119895 le

119899119887 minus 1 Let 119904 ge 0 be minimal such that (119904 119887) isin 119867Furthermore consider the natural inclusion 120580

1 Z119898

rarr

119866 given by 1205801(119909) = (119909 0)Then 120580minus1

1(119867) is a subgroup ofZ

119898and

there exists a unique divisor 119886 of119898 such that 120580minus11(119867) = ⟨119886⟩

We show that 119867 = (119894119886 + 119895119904 119895119887) 119894 119895 isin Z Indeed forevery 119894 119895 isin Z (119894119886+119895119904 119895119887) = 119894(119886 0)+119895(119904 119887) isin 119867 On the otherhand for every (119906 V) isin 119867 one has V isin 120587

2(119867) and hence there

is 119895 isin Z such that V = 119895119887 We obtain (119906 minus 119895119904 0) = (119906 V) minus119895(119904 119887) isin 119867 119906minus119895119904 isin 120580

minus1

1(119867) and there is 119894 isin Zwith 119906minus119895119904 = 119894119886

Here a necessary condition is that (119904119899119887 0) isin 119867 (obtainedfor 119894 = 0 and 119895 = 119899119887) that is 119886 | 119904119899b equivalent to119886gcd(119886 119899119887) | 119904 Clearly if this is verified then for the aboverepresentation of 119867 it is enough to take the values 0 le 119894 le

119898119886 minus 1 and 0 le 119895 le 119899119887 minus 1Also dividing 119904 by 119886 we have 119904 = 119886119902 + 119903 with 0 le 119903 lt 119886

and (119903 119887) = (119904 119887) minus 119902(119886 0) isin 119867 showing that 119904 lt 119886 byits minimality Hence 119904 = 119905119886gcd(119886 119899119887) with 0 le 119905 le

gcd(119886 119899119887) minus 1 Thus we obtain the given representationConversely every (119886 119887 119905) isin 119868

119898119899generates a subgroup

119867119886119887119905

of order 119898119899(119886119887) of Z119898

times Z119899and the proof is

complete

Proof of Theorem 2 (i)-(ii) We first determine the exponentof the subgroup 119867

119886119887119905 119867119886119887119905

is generated by (119886 0) and (119904 119887)hence its exponent is the least commonmultiple of the ordersof these two elements The order of (119886 0) is119898119886 To computethe order of (119904 119887) note that119898 | 119903119904 if and only if119898gcd(119898 119904) |

119903Thus the order of (119904 119887) is lcm(119898gcd(119898 119904) 119899119887)We deducethat the exponent of119867

119886119887119905is

lcm(

119898

119886

119898

gcd (119898 119904)

119899

119887

)

= lcm(

119898119899

119899119886

119898119899

119899gcd (119898 119904)

119898119899

119898119887

)

=

119898119899

gcd (119899119886 119899119898 119899119904 119898119887)

=

119898119899

gcd (119898119887 119899119886 119899119904)

= 120573

(27)

For every finite Abelian group the rank of a nontrivialsubgroup is at most the rank of the groupTherefore the rankof 119867119886119887t is 1 or 2 That is 119867

119886119887119905≃ Z119860times Z119861with certain

119860 119861 isin N such that 119860 | 119861 Here the exponent of 119867119886119887119905

equalsthat of Z

119860times Z119861 which is lcm(119860 119861) = 119861 Using (ii) already

proved we deduce that 119861 = 120573 Since the order of 119867119886119887119905

is119860119861 = 119898119899(119886119887) we have 119860 = 119898119899(119886119887120573) = 120572

(iii) According to (i) 119867119886119887119905

≃ Z120572timesZ120573 where 120572 | 120573 Hence

119867119886119887119905

is cyclic if and only if 120572 = 1(iv) The subgroups of Z

119898are of form 119894119886 0 le 119894 le 119898119886 minus

1 where 119886 | 119898 and the properties follow from (6) and (iii)

Proof of Theorem 3 By its definition the number of elementsof the set 119868

119898119899is

sum

119886|119898119887|119899

sum

0le119905legcd(119886119899119887)minus11

= sum

119886|119898119887|119899

gcd(119886 119899119887

) = sum

119886|119898119887|119899

gcd (119886 119887) (28)

representing 119904(119898 119899) This is formula (10)To obtain formula (11) apply the Gauss formula 119899 =

sum119889|119899

120601(119889) (119899 isin N) by writing the following

119904 (119898 119899) = sum

119886|119898119887|119899

sum

119889|gcd(119886119887)120601 (119889) = sum

119886119909=119898

119887119910=119899

sum

119889119894=119886

119889119895=119887

120601 (119889) = sum

119889119894119909=119898

119889119895119910=119899

120601 (119889)

= sum

119889119906=119898

119889V=119899

120601 (119889) sum

119894119909=119906

119895119910=V

1 = sum

119889119906=119898

119889V=119899

120601 (119889) 120591 (119906) 120591 (V)

= sum

119889|gcd(119898119899)120601 (119889) 120591 (

119898

119889

) 120591 (

119899

119889

)

(29)

Journal of Numbers 5

Now (12) follows from (11) by the Busche-Ramanujanidentity (cf [19 Chapter 1])

120591 (119898) 120591 (119899) = sum

119889|gcd(119898119899)120591 (

119898119899

1198892) (119898 119899 isin N) (30)

Proof of Theorem 4 According toTheorem 1

119904119896(119898 119899) = sum

119886|119898119887|119899

119898119899119886119887=119896

gcd(119886 119899119887

) (31)

giving (13) which can be written by Gaussrsquo formula again as

119904119896(119898 119899) = sum

119886|119898119887|119899

119898119887119886=119896

sum

119888|119886119888|119887

120601 (119888) = sum

119888119894119909=119898

119888119895119910=119899

119888119895119909=119896

120601 (119888)

= sum

119889119894=119898

119890119910=119899

sum

119888119909=119889

119888119895=119890

119888119895119909=119896

120601 (119888)

(32)

where in the inner sum one has 119888 = 119889119890119896 and obtains (14)Now to get (15) write (32) as

119904119896(119898 119899) = sum

119888119906=119898

119888V=119899119888119908=119896

120601 (119888) sum

119894119909=119906

119895119910=V119895119909=119908

1 = sum

119888119906=119898

119888V=119899119888119908=119896

120601 (119888)119873 (119906 V 119908)

= sum

119888|gcd(119898119899119896)120601 (119888)119873(

119898

119888

119899

119888

119896

119888

)

(33)

and the proof is complete

Proof of Theorem 5 (i) According to Theorems 1 and 2(iii)and using that sum

119889|119899120583(119889) = 1 or 0 and according to 119899 = 1 or

119899 gt 1

119888 (119898 119899) = sum

119886|119898119887|119899

sum

1le119904le119886

119886119887|119899119904

gcd(119898119886119899119887119899119904119886119887)=1

1 = sum

119886119909=119898

119887119910=119899

sum

1le119904le119886

119886119903=119910119904

gcd(119909119910119903)=1

1

= sum

119886119909=119898

119887119910=119899

sum

1le119904le119886

119886119903=119910119904

sum

119890|gcd(119909119910119903)

120583 (119890) = sum

119886119890119894=119898

119887119890119895=119899

120583 (119890) sum

1le119904le119886

119886gcd(119886119895)|119904

1

(34)

where the inner sum is gcd(119886 119895) Hence

119888 (119898 119899) = sum

119886119890119894=119898

119887119890119895=119899

120583 (119890) gcd (119886 119895) (35)

Now regrouping the terms according to 119890119894 = 119911 and 119887119890 = 119905

we obtain

119888 (119898 119899) = sum

119886119911=119898

119895119905=119899

gcd (119886 119895) sum119890119894=119911

119887119890=119905

120583 (119890) = sum

119886119911=119898

119895119905=119899

gcd (119886 119895) sum

119890|gcd(119911119905)120583 (119890)

= sum

119886119911=119898

119895119905=119899

gcd(119911119905)=1

gcd (119886 119895)

(36)

which is (16)

The next results follow applying Gaussrsquo formula andthe Busche-Ramanujan formula similar to the proof ofTheorem 3

(ii) For the subproducts 1198671198861198870

the values 119886 | 119898 and 119887 | 119899

can be chosen arbitrary and it follows at once that the numberof subproducts is 120591(119898)120591(119899)Thenumber of cyclic subproductsis

sum

119886|119898

119887|119899

gcd(119898119886119899119887)=1

1 = sum

119886119909=119898

119887119910=119899

gcd(119909119910)=1

1 = sum

119886119909=119898

119887119910=119899

sum

119890|gcd(119909119910)

120583 (119890)

= sum

119890119860=119898

119890119861=119899

120583 (119890) 120591 (119860) 120591 (119861)

= sum

119890|gcd(119898119899)120583 (119890) 120591 (

119898

119890

) 120591 (

119899

119890

) = 120591 (119898119899)

(37)

by the inverse Busche-Ramanujan identity

Proof of Theorem 8 According to Theorem 2(ii) the num-ber of subgroups of exponent 120575 of Z

119899times Z119899is

119864120575(119899) = sum

119886|119899119887|119899

sum

1le119904le119886

119886119887|119899119904

119899gcd(119886119887119904)=120575

1 = sum

119886119909=119899

119887119910=119899

sum

1le119904le119886

119886119903=119910119904

gcd(119886119887119904)=119899120575

1

(38)

Write 119886 = 1198861119899120575 119887 = 119887

1119899120575 and 119904 = 119904

1119899120575 with

gcd(1198861 1198871 1199041) = 1 We deduce similar to the proof of

Theorem 5(i) that

119864120575(119899) = sum

119890119894119909=120575

119890119895119910=120575

120583 (119890) gcd (119894 119910) (39)

which is exactly 119888(120575 120575) = 119888(120575) (cf (35))

5 Further Remarks

(1) As mentioned in Section 2 the functions (119898 119899) 997891rarr

119904(119898 119899) and (119898 119899) 997891rarr 119888(119898 119899) are multiplicative func-tions of two variablesThis follows easily from formu-lae (10) and (17) respectively Namely according tothose formulae 119904(119898 119899) and 119888(119898 119899) are two variablesDirichlet convolutions of the functions (119898 119899) 997891rarr

gcd(119898 119899) and (119898 119899) 997891rarr 120601(gcd(119898 119899)) respectivelywith the constant 1 function all multiplicative Sinceconvolution preserves the multiplicativity we deducethat 119904(119898 119899) and 119888(119898 119899) are alsomultiplicative See [17Section 2] for details

(2) Asymptotic formulas with sharp error terms for thesums sum

119898119899le119909119904(119898 119899) and sum

119898119899le119909119888(119898 119899) were given in

the paper [20](3) For any finite groups 119860 and 119861 a subgroup 119862 of 119860 times 119861

is cyclic if and only if 120580minus11(119862) and 120580

minus1

2(119862) have coprime

orders where 1205801and 1205802are the natural inclusions ([21

Theorem 42]) In the case 119860 = Z119898 119861 = Z

119899 and

119862 = 119867119886119887119905

one has 120580minus11(119862) = 119898119886 and 120580minus1

2(119862) =

gcd(119899119887 119899119904119886119887) and the characterization of the cyclicsubgroups 119867

119886119887119905given in Theorem 2(iii) can be

6 Journal of Numbers

obtained also in this way It turns out that regardingthe sublattice 119867

119886119887119905is cyclic if and only if the

numbers of points on the horizontal and vertical axesrespectively are relatively prime Note that in the case119898 = 119899 the above condition reads 119899gcd(119886 119887 119904) = 119886119887Thus it is necessary that 119899 | 119886119887 The subgroup onFigure 1 is not cyclic

(4) Note also the next formula for the number of cyclicsubgroups of Z

119899times Z119899 derived in [22 Example 2]

119888 (119899) = sum

lcm(119889119890)=119899gcd (119889 119890) (119899 isin N) (40)

where the sum is over all ordered pairs (119889 119890) such thatlcm(119889 119890) = 119899 For a short direct proof of (40) write119889 = ℓ119886 119890 = ℓ119887 with gcd(119886 119887) = 1 Then gcd(119889 119890) = ℓlcm(119889 119890) = ℓ119886119887 and obtain

sum

lcm(119889119890)=119899gcd (119889 119890)

= sum

ℓ119886119887=119899

gcd(119886119887)=1

ℓ = sum

ℓ119896=119899

ℓ sum

119886119887=119896

gcd(119886119887)=1

1 = sum

ℓ119896=119899

ℓ2120596(119896)

= 119888 (119899)

(41)

according to (21)(5) Every subgroup119870 ofZtimesZ has the representation119870 =

(119894119886 + 119895119904 119895119887) 119894 119895 isin Z where 0 le 119904 le 119886 and 0 le 119887

are unique integers This follows like in the proof ofTheorem 1 Furthermore in the case 119886 119887 ge 1 0 le 119904 le

119886minus 1 the index of119870 is 119886119887 and one obtains at once thatthe number of subgroups 119870 having index 119899 (119899 isin N)is sum119886119887=119899

sum0le119904le119886minus1

1 = sum119886119887=119899

119886 = 120590(119899) mentioned inSection 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Nicki Holighaus was partially supported by the AustrianScience Fund (FWF) START-Project FLAME (Y551-N13)Laszlo Toth gratefully acknowledges support from the Aus-trian Science Fund (FWF) under Project no M1376-N18Christoph Wiesmeyr was partially supported by EU FETOpen Grant UNLocX (255931)

References

[1] M J Grady ldquoA group theoretic approach to a famous partitionformulardquo The American Mathematical Monthly vol 112 no 7pp 645ndash651 2005

[2] Y M Zou ldquoGaussian binomials and the number of sublatticesrdquoActa Crystallographica vol 62 no 5 pp 409ndash410 2006

[3] ldquoThe On-Line Encyclopedia of Integer Sequencesrdquo httpoeisorg

[4] K Grochenig Foundations of Time-Frequency Analysis Appliedand Numerical Harmonic Analysis Birkhauser Boston MassUSA 2001

[5] G Kutyniok and T Strohmer ldquoWilson bases for general time-frequency latticesrdquo SIAM Journal onMathematical Analysis vol37 no 3 pp 685ndash711 (electronic) 2005

[6] A J van Leest Non-separable Gabor schemes their design andimplementation [PhD thesis] Technische Universiteit Eind-hoven Eindhoven The Netherlands 2001

[7] T Strohmer ldquoNumerical algorithms for discrete Gabor expan-sionsrdquo in Gabor Analysis and Algorithms Theory and Applica-tions H G Feichtinger and T Strohmer Eds pp 267ndash294Birkhauser Boston Mass USA 1998

[8] A Machı Groups An Introduction to Ideas and Methods of theTheory of Groups Springer 2012

[9] J J Rotman An Introduction to theTheory of Groups vol 148 ofGraduate Texts in Mathematics Springer New York NY USA4th edition 1995

[10] R Schmidt Subgroup Lattices of Groups deGruyter Expositionsin Mathematics 14 de Gruyter Berlin Germany 1994

[11] M Suzuki ldquoOn the lattice of subgroups of finite groupsrdquoTransactions of the American Mathematical Society vol 70 pp345ndash371 1951

[12] G Calugareanu ldquoThe total number of subgroups of a finiteabelian grouprdquo Scientiae Mathematicae Japonicae vol 60 no1 pp 157ndash167 2004

[13] J Petrillo ldquoCounting subgroups in a direct product of finitecyclic groupsrdquo College Mathematics Journal vol 42 no 3 pp215ndash222 2011

[14] M Tarnauceanu ldquoA new method of proving some classicaltheorems of abelian groupsrdquo Southeast Asian Bulletin of Mathe-matics vol 31 no 6 pp 1191ndash1203 2007

[15] M Tarnauceanu ldquoAn arithmetic method of counting the sub-groups of a finite abelian grouprdquo Bulletin Mathematiques de laSociete des SciencesMathematiques de Roumanie vol 53 no 101pp 373ndash386 2010

[16] W C Calhoun ldquoCounting the subgroups of some finite groupsrdquoThe American Mathematical Monthly vol 94 no 1 pp 54ndash591987

[17] L Toth ldquoMenons identity and arithmetical sums representingfunctions of several variablesrdquoRendiconti del SeminarioMatem-atico Universita e Politecnico di Torino vol 69 no 1 pp 97ndash1102011

[18] L Toth ldquoOn the number of cyclic subgroups of a finite Abeliangrouprdquo Bulletin Mathematique de la Societe des Sciences Mathe-matiques de Roumanie vol 55 no 103 pp 423ndash428 2012

[19] P J McCarthy Introduction to Arithmetical Functions SpringerNew York NY USA 1986

[20] WGNowak andL Toth ldquoOn the average number of subgroupsof the groupZ

119898timesZ119899rdquo International Journal of NumberTheory

vol 10 pp 363ndash374 2014[21] K Bauer D Sen and P Zvengrowski ldquoA generalized Goursat

lemmardquo httparxivorgabs11090024[22] A Pakapongpun and T Ward ldquoFunctorial orbit countingrdquo

Journal of Integer Sequences vol 12 Article ID 0924 20 pages2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Numbers 5

Now (12) follows from (11) by the Busche-Ramanujanidentity (cf [19 Chapter 1])

120591 (119898) 120591 (119899) = sum

119889|gcd(119898119899)120591 (

119898119899

1198892) (119898 119899 isin N) (30)

Proof of Theorem 4 According toTheorem 1

119904119896(119898 119899) = sum

119886|119898119887|119899

119898119899119886119887=119896

gcd(119886 119899119887

) (31)

giving (13) which can be written by Gaussrsquo formula again as

119904119896(119898 119899) = sum

119886|119898119887|119899

119898119887119886=119896

sum

119888|119886119888|119887

120601 (119888) = sum

119888119894119909=119898

119888119895119910=119899

119888119895119909=119896

120601 (119888)

= sum

119889119894=119898

119890119910=119899

sum

119888119909=119889

119888119895=119890

119888119895119909=119896

120601 (119888)

(32)

where in the inner sum one has 119888 = 119889119890119896 and obtains (14)Now to get (15) write (32) as

119904119896(119898 119899) = sum

119888119906=119898

119888V=119899119888119908=119896

120601 (119888) sum

119894119909=119906

119895119910=V119895119909=119908

1 = sum

119888119906=119898

119888V=119899119888119908=119896

120601 (119888)119873 (119906 V 119908)

= sum

119888|gcd(119898119899119896)120601 (119888)119873(

119898

119888

119899

119888

119896

119888

)

(33)

and the proof is complete

Proof of Theorem 5 (i) According to Theorems 1 and 2(iii)and using that sum

119889|119899120583(119889) = 1 or 0 and according to 119899 = 1 or

119899 gt 1

119888 (119898 119899) = sum

119886|119898119887|119899

sum

1le119904le119886

119886119887|119899119904

gcd(119898119886119899119887119899119904119886119887)=1

1 = sum

119886119909=119898

119887119910=119899

sum

1le119904le119886

119886119903=119910119904

gcd(119909119910119903)=1

1

= sum

119886119909=119898

119887119910=119899

sum

1le119904le119886

119886119903=119910119904

sum

119890|gcd(119909119910119903)

120583 (119890) = sum

119886119890119894=119898

119887119890119895=119899

120583 (119890) sum

1le119904le119886

119886gcd(119886119895)|119904

1

(34)

where the inner sum is gcd(119886 119895) Hence

119888 (119898 119899) = sum

119886119890119894=119898

119887119890119895=119899

120583 (119890) gcd (119886 119895) (35)

Now regrouping the terms according to 119890119894 = 119911 and 119887119890 = 119905

we obtain

119888 (119898 119899) = sum

119886119911=119898

119895119905=119899

gcd (119886 119895) sum119890119894=119911

119887119890=119905

120583 (119890) = sum

119886119911=119898

119895119905=119899

gcd (119886 119895) sum

119890|gcd(119911119905)120583 (119890)

= sum

119886119911=119898

119895119905=119899

gcd(119911119905)=1

gcd (119886 119895)

(36)

which is (16)

The next results follow applying Gaussrsquo formula andthe Busche-Ramanujan formula similar to the proof ofTheorem 3

(ii) For the subproducts 1198671198861198870

the values 119886 | 119898 and 119887 | 119899

can be chosen arbitrary and it follows at once that the numberof subproducts is 120591(119898)120591(119899)Thenumber of cyclic subproductsis

sum

119886|119898

119887|119899

gcd(119898119886119899119887)=1

1 = sum

119886119909=119898

119887119910=119899

gcd(119909119910)=1

1 = sum

119886119909=119898

119887119910=119899

sum

119890|gcd(119909119910)

120583 (119890)

= sum

119890119860=119898

119890119861=119899

120583 (119890) 120591 (119860) 120591 (119861)

= sum

119890|gcd(119898119899)120583 (119890) 120591 (

119898

119890

) 120591 (

119899

119890

) = 120591 (119898119899)

(37)

by the inverse Busche-Ramanujan identity

Proof of Theorem 8 According to Theorem 2(ii) the num-ber of subgroups of exponent 120575 of Z

119899times Z119899is

119864120575(119899) = sum

119886|119899119887|119899

sum

1le119904le119886

119886119887|119899119904

119899gcd(119886119887119904)=120575

1 = sum

119886119909=119899

119887119910=119899

sum

1le119904le119886

119886119903=119910119904

gcd(119886119887119904)=119899120575

1

(38)

Write 119886 = 1198861119899120575 119887 = 119887

1119899120575 and 119904 = 119904

1119899120575 with

gcd(1198861 1198871 1199041) = 1 We deduce similar to the proof of

Theorem 5(i) that

119864120575(119899) = sum

119890119894119909=120575

119890119895119910=120575

120583 (119890) gcd (119894 119910) (39)

which is exactly 119888(120575 120575) = 119888(120575) (cf (35))

5 Further Remarks

(1) As mentioned in Section 2 the functions (119898 119899) 997891rarr

119904(119898 119899) and (119898 119899) 997891rarr 119888(119898 119899) are multiplicative func-tions of two variablesThis follows easily from formu-lae (10) and (17) respectively Namely according tothose formulae 119904(119898 119899) and 119888(119898 119899) are two variablesDirichlet convolutions of the functions (119898 119899) 997891rarr

gcd(119898 119899) and (119898 119899) 997891rarr 120601(gcd(119898 119899)) respectivelywith the constant 1 function all multiplicative Sinceconvolution preserves the multiplicativity we deducethat 119904(119898 119899) and 119888(119898 119899) are alsomultiplicative See [17Section 2] for details

(2) Asymptotic formulas with sharp error terms for thesums sum

119898119899le119909119904(119898 119899) and sum

119898119899le119909119888(119898 119899) were given in

the paper [20](3) For any finite groups 119860 and 119861 a subgroup 119862 of 119860 times 119861

is cyclic if and only if 120580minus11(119862) and 120580

minus1

2(119862) have coprime

orders where 1205801and 1205802are the natural inclusions ([21

Theorem 42]) In the case 119860 = Z119898 119861 = Z

119899 and

119862 = 119867119886119887119905

one has 120580minus11(119862) = 119898119886 and 120580minus1

2(119862) =

gcd(119899119887 119899119904119886119887) and the characterization of the cyclicsubgroups 119867

119886119887119905given in Theorem 2(iii) can be

6 Journal of Numbers

obtained also in this way It turns out that regardingthe sublattice 119867

119886119887119905is cyclic if and only if the

numbers of points on the horizontal and vertical axesrespectively are relatively prime Note that in the case119898 = 119899 the above condition reads 119899gcd(119886 119887 119904) = 119886119887Thus it is necessary that 119899 | 119886119887 The subgroup onFigure 1 is not cyclic

(4) Note also the next formula for the number of cyclicsubgroups of Z

119899times Z119899 derived in [22 Example 2]

119888 (119899) = sum

lcm(119889119890)=119899gcd (119889 119890) (119899 isin N) (40)

where the sum is over all ordered pairs (119889 119890) such thatlcm(119889 119890) = 119899 For a short direct proof of (40) write119889 = ℓ119886 119890 = ℓ119887 with gcd(119886 119887) = 1 Then gcd(119889 119890) = ℓlcm(119889 119890) = ℓ119886119887 and obtain

sum

lcm(119889119890)=119899gcd (119889 119890)

= sum

ℓ119886119887=119899

gcd(119886119887)=1

ℓ = sum

ℓ119896=119899

ℓ sum

119886119887=119896

gcd(119886119887)=1

1 = sum

ℓ119896=119899

ℓ2120596(119896)

= 119888 (119899)

(41)

according to (21)(5) Every subgroup119870 ofZtimesZ has the representation119870 =

(119894119886 + 119895119904 119895119887) 119894 119895 isin Z where 0 le 119904 le 119886 and 0 le 119887

are unique integers This follows like in the proof ofTheorem 1 Furthermore in the case 119886 119887 ge 1 0 le 119904 le

119886minus 1 the index of119870 is 119886119887 and one obtains at once thatthe number of subgroups 119870 having index 119899 (119899 isin N)is sum119886119887=119899

sum0le119904le119886minus1

1 = sum119886119887=119899

119886 = 120590(119899) mentioned inSection 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Nicki Holighaus was partially supported by the AustrianScience Fund (FWF) START-Project FLAME (Y551-N13)Laszlo Toth gratefully acknowledges support from the Aus-trian Science Fund (FWF) under Project no M1376-N18Christoph Wiesmeyr was partially supported by EU FETOpen Grant UNLocX (255931)

References

[1] M J Grady ldquoA group theoretic approach to a famous partitionformulardquo The American Mathematical Monthly vol 112 no 7pp 645ndash651 2005

[2] Y M Zou ldquoGaussian binomials and the number of sublatticesrdquoActa Crystallographica vol 62 no 5 pp 409ndash410 2006

[3] ldquoThe On-Line Encyclopedia of Integer Sequencesrdquo httpoeisorg

[4] K Grochenig Foundations of Time-Frequency Analysis Appliedand Numerical Harmonic Analysis Birkhauser Boston MassUSA 2001

[5] G Kutyniok and T Strohmer ldquoWilson bases for general time-frequency latticesrdquo SIAM Journal onMathematical Analysis vol37 no 3 pp 685ndash711 (electronic) 2005

[6] A J van Leest Non-separable Gabor schemes their design andimplementation [PhD thesis] Technische Universiteit Eind-hoven Eindhoven The Netherlands 2001

[7] T Strohmer ldquoNumerical algorithms for discrete Gabor expan-sionsrdquo in Gabor Analysis and Algorithms Theory and Applica-tions H G Feichtinger and T Strohmer Eds pp 267ndash294Birkhauser Boston Mass USA 1998

[8] A Machı Groups An Introduction to Ideas and Methods of theTheory of Groups Springer 2012

[9] J J Rotman An Introduction to theTheory of Groups vol 148 ofGraduate Texts in Mathematics Springer New York NY USA4th edition 1995

[10] R Schmidt Subgroup Lattices of Groups deGruyter Expositionsin Mathematics 14 de Gruyter Berlin Germany 1994

[11] M Suzuki ldquoOn the lattice of subgroups of finite groupsrdquoTransactions of the American Mathematical Society vol 70 pp345ndash371 1951

[12] G Calugareanu ldquoThe total number of subgroups of a finiteabelian grouprdquo Scientiae Mathematicae Japonicae vol 60 no1 pp 157ndash167 2004

[13] J Petrillo ldquoCounting subgroups in a direct product of finitecyclic groupsrdquo College Mathematics Journal vol 42 no 3 pp215ndash222 2011

[14] M Tarnauceanu ldquoA new method of proving some classicaltheorems of abelian groupsrdquo Southeast Asian Bulletin of Mathe-matics vol 31 no 6 pp 1191ndash1203 2007

[15] M Tarnauceanu ldquoAn arithmetic method of counting the sub-groups of a finite abelian grouprdquo Bulletin Mathematiques de laSociete des SciencesMathematiques de Roumanie vol 53 no 101pp 373ndash386 2010

[16] W C Calhoun ldquoCounting the subgroups of some finite groupsrdquoThe American Mathematical Monthly vol 94 no 1 pp 54ndash591987

[17] L Toth ldquoMenons identity and arithmetical sums representingfunctions of several variablesrdquoRendiconti del SeminarioMatem-atico Universita e Politecnico di Torino vol 69 no 1 pp 97ndash1102011

[18] L Toth ldquoOn the number of cyclic subgroups of a finite Abeliangrouprdquo Bulletin Mathematique de la Societe des Sciences Mathe-matiques de Roumanie vol 55 no 103 pp 423ndash428 2012

[19] P J McCarthy Introduction to Arithmetical Functions SpringerNew York NY USA 1986

[20] WGNowak andL Toth ldquoOn the average number of subgroupsof the groupZ

119898timesZ119899rdquo International Journal of NumberTheory

vol 10 pp 363ndash374 2014[21] K Bauer D Sen and P Zvengrowski ldquoA generalized Goursat

lemmardquo httparxivorgabs11090024[22] A Pakapongpun and T Ward ldquoFunctorial orbit countingrdquo

Journal of Integer Sequences vol 12 Article ID 0924 20 pages2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Numbers

obtained also in this way It turns out that regardingthe sublattice 119867

119886119887119905is cyclic if and only if the

numbers of points on the horizontal and vertical axesrespectively are relatively prime Note that in the case119898 = 119899 the above condition reads 119899gcd(119886 119887 119904) = 119886119887Thus it is necessary that 119899 | 119886119887 The subgroup onFigure 1 is not cyclic

(4) Note also the next formula for the number of cyclicsubgroups of Z

119899times Z119899 derived in [22 Example 2]

119888 (119899) = sum

lcm(119889119890)=119899gcd (119889 119890) (119899 isin N) (40)

where the sum is over all ordered pairs (119889 119890) such thatlcm(119889 119890) = 119899 For a short direct proof of (40) write119889 = ℓ119886 119890 = ℓ119887 with gcd(119886 119887) = 1 Then gcd(119889 119890) = ℓlcm(119889 119890) = ℓ119886119887 and obtain

sum

lcm(119889119890)=119899gcd (119889 119890)

= sum

ℓ119886119887=119899

gcd(119886119887)=1

ℓ = sum

ℓ119896=119899

ℓ sum

119886119887=119896

gcd(119886119887)=1

1 = sum

ℓ119896=119899

ℓ2120596(119896)

= 119888 (119899)

(41)

according to (21)(5) Every subgroup119870 ofZtimesZ has the representation119870 =

(119894119886 + 119895119904 119895119887) 119894 119895 isin Z where 0 le 119904 le 119886 and 0 le 119887

are unique integers This follows like in the proof ofTheorem 1 Furthermore in the case 119886 119887 ge 1 0 le 119904 le

119886minus 1 the index of119870 is 119886119887 and one obtains at once thatthe number of subgroups 119870 having index 119899 (119899 isin N)is sum119886119887=119899

sum0le119904le119886minus1

1 = sum119886119887=119899

119886 = 120590(119899) mentioned inSection 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Nicki Holighaus was partially supported by the AustrianScience Fund (FWF) START-Project FLAME (Y551-N13)Laszlo Toth gratefully acknowledges support from the Aus-trian Science Fund (FWF) under Project no M1376-N18Christoph Wiesmeyr was partially supported by EU FETOpen Grant UNLocX (255931)

References

[1] M J Grady ldquoA group theoretic approach to a famous partitionformulardquo The American Mathematical Monthly vol 112 no 7pp 645ndash651 2005

[2] Y M Zou ldquoGaussian binomials and the number of sublatticesrdquoActa Crystallographica vol 62 no 5 pp 409ndash410 2006

[3] ldquoThe On-Line Encyclopedia of Integer Sequencesrdquo httpoeisorg

[4] K Grochenig Foundations of Time-Frequency Analysis Appliedand Numerical Harmonic Analysis Birkhauser Boston MassUSA 2001

[5] G Kutyniok and T Strohmer ldquoWilson bases for general time-frequency latticesrdquo SIAM Journal onMathematical Analysis vol37 no 3 pp 685ndash711 (electronic) 2005

[6] A J van Leest Non-separable Gabor schemes their design andimplementation [PhD thesis] Technische Universiteit Eind-hoven Eindhoven The Netherlands 2001

[7] T Strohmer ldquoNumerical algorithms for discrete Gabor expan-sionsrdquo in Gabor Analysis and Algorithms Theory and Applica-tions H G Feichtinger and T Strohmer Eds pp 267ndash294Birkhauser Boston Mass USA 1998

[8] A Machı Groups An Introduction to Ideas and Methods of theTheory of Groups Springer 2012

[9] J J Rotman An Introduction to theTheory of Groups vol 148 ofGraduate Texts in Mathematics Springer New York NY USA4th edition 1995

[10] R Schmidt Subgroup Lattices of Groups deGruyter Expositionsin Mathematics 14 de Gruyter Berlin Germany 1994

[11] M Suzuki ldquoOn the lattice of subgroups of finite groupsrdquoTransactions of the American Mathematical Society vol 70 pp345ndash371 1951

[12] G Calugareanu ldquoThe total number of subgroups of a finiteabelian grouprdquo Scientiae Mathematicae Japonicae vol 60 no1 pp 157ndash167 2004

[13] J Petrillo ldquoCounting subgroups in a direct product of finitecyclic groupsrdquo College Mathematics Journal vol 42 no 3 pp215ndash222 2011

[14] M Tarnauceanu ldquoA new method of proving some classicaltheorems of abelian groupsrdquo Southeast Asian Bulletin of Mathe-matics vol 31 no 6 pp 1191ndash1203 2007

[15] M Tarnauceanu ldquoAn arithmetic method of counting the sub-groups of a finite abelian grouprdquo Bulletin Mathematiques de laSociete des SciencesMathematiques de Roumanie vol 53 no 101pp 373ndash386 2010

[16] W C Calhoun ldquoCounting the subgroups of some finite groupsrdquoThe American Mathematical Monthly vol 94 no 1 pp 54ndash591987

[17] L Toth ldquoMenons identity and arithmetical sums representingfunctions of several variablesrdquoRendiconti del SeminarioMatem-atico Universita e Politecnico di Torino vol 69 no 1 pp 97ndash1102011

[18] L Toth ldquoOn the number of cyclic subgroups of a finite Abeliangrouprdquo Bulletin Mathematique de la Societe des Sciences Mathe-matiques de Roumanie vol 55 no 103 pp 423ndash428 2012

[19] P J McCarthy Introduction to Arithmetical Functions SpringerNew York NY USA 1986

[20] WGNowak andL Toth ldquoOn the average number of subgroupsof the groupZ

119898timesZ119899rdquo International Journal of NumberTheory

vol 10 pp 363ndash374 2014[21] K Bauer D Sen and P Zvengrowski ldquoA generalized Goursat

lemmardquo httparxivorgabs11090024[22] A Pakapongpun and T Ward ldquoFunctorial orbit countingrdquo

Journal of Integer Sequences vol 12 Article ID 0924 20 pages2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of