A partition formula connected with Abelian groups B y P . HALL, C a m b r i d g e ( E n g l a n d )

Let p be a given prime. The object of this note is to prove the following ra ther curious result.

The sum o] the reciprocals o/ the orders o/ all the Abelian groups of order a Tower o] p is equal to the sum o] the reciprocals o/ the orders o~ their groups o/ automorphisms.

I t is well known t h a t the Abelian groups of order pn s tand in (1 - - 1) correspondence with the eo (n) unrestr icted par t i t ions of n, the par t i t ion corresponding to a given Abelian group being called its type.

Thus the sum of the reciprocals of the orders of all the Abelian groups of order a power of p is equal to

~ (n) (1)

Writ ing ,=0 p~ l,,(x) ~ (1 - - x) ( 1 - - x 2) ( 1 - - x3) . . . ( 1 - - x n) , /o(x) ---- 1 , (2)

and 1 e = - - , (3)

P the value of the sum (1) is easily seen to be

1 l| (Q) (4)

But , by an iden t i ty due to Euler, this is the same as

~n " (5)

And the theorem ment ioned above will accordingly follow once we have shown t h a t the sum o/ the reciprocals o~ the orders o/ the groups o/auto- morphisms o/ the co (n) Abelian groups o] order p'~ is equal to ~nl /~ (~).

For part i t ions we use the no ta t ion of Maemahon. Thus, the Abelian group G of order pn and type

(1 z, 2 z= 3z* . . . ) (6)

is the direct product of cyclic groups, 41 of which are of order p, ~ of order p~, h a of order p3, and so on. Clearly,

n = ~t, -}- 2 ;t 2 -l- 3 ~t 3 + "" . (7)

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The par t i t ion of n which is associated with the par t i t ion (6) has the parts #1,/~2, ..- given by

#, = 4, q- Ai+, q- Xi+2 q- "'" (i -= 1, 2 , . . . ) . (S)

Thus, since h i > / 0 for each i, we have

#1 /> /g2 /> "'" >/ 0 . (9)

And plainly, from (7) and (8),

~tl -4- #2 + . . . . n . (10)

Conversely, given any par t i t ion of n in the form (9), (10), the associated par t i t ion (6) is obtained a t once by the rule t h a t

2~, = Ix,--I~,+l (i = 1, 2, . . .) . (11)

The associated par t i t ion has a simple meaning for the group G. Let Gk denote the characteristic subgroup of G which consists of all elements of O of order pk or less. Then

1 - - - G o < G I < G 2 < . . . < G m = G ,

where m is the largest of the type invar iants 1) of (7, and the order o/ Gk l G~_ 1 is precisely Ir "~.

Now the order of the group of automorphisms of G can be expressed very simply in terms of the "associated invar ian ts" g~. I t is 2)

f t t , - g . (e) f g , - g , (e) . - . (12) Qgl gz + "'"

And the result we require to prove is the ease x = e of the ident i ty

s

Xn X g l + g ~ + . . . - - X (13) /~ (x) (~)/~,_~(x)/~_~ ( x ) . . . '

the sum being t aken over all oJ (n) part i t ions (9), (10) of the number n.

The various terms of (13) m a y be regarded as the generat ing funct ions of par t i t ions or compositions of certain definite kinds. For example, the coefficient of ~ on the left of (13) is equal to the number of part i t ions of N for which the greatest par t is exact ly n. As a first step in the proof of the ident i ty , we shall connect every such par t i t ion of N wi th a part icular

1) I . e . ~ n > 0 , ~m+l = kin+2 . . . . . 0 . ~) Cf. A. Spelser, T h e o r i e d e r G r u p p e n v o n e n d l i c h e r O r d n u n g , 3or.

Aufl., w 43, Sa t z 114.

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o n e of the r (n) partitions (9), (10) of n, and thereby with a particular one of the co (n) summands on the right of (13).

This may be done most conveniently by means of the graph 3) of the partition of N in question. Let the parts of this partition, arranged in descending order of magnitude be N1, N~ . . . . , so tha t we have

(14) N 1 --}- Nu -~- . . . . N �9

Then its graph may be defined to consist of a set of N coplanar lattice- points, viz. all those points whose Cartesian coordinates (x, y) are positive integers satisfying

x ~< N , . (15)

(When y exceeds the number of parts of (14), we take N~ ---- 0.) We are now able to define, successively, the numbers #1, / .2 , . . . . .

which correspond to the partition (14). We take /.1 to be the greatest integer such tha t the point (/~1,/*1)

belongs to the graph (15). Next, supposing tha t #1, #~, . . . , #i-1 have already been defined, and tha t their sum is less than n, we define/*i to be the greatest integer such tha t (/.1 + #3 + "'" -~/~i, #i) is a point of the graph.

I t follows at once, from (14) and (15), tha t the numbers #i so defined satisfy (9) and (10). Plainly, also, the square of #~ lattice-points having for opposite corners the points (/'1 + "'" ~-/~-1 ~- 1, 1) and (/~1 ~- "'" #~, #~) belongs entirely to the graph. Thus, ff we write

M -= N - - #~ - - / ,~ . . . . , (16)

there remain, outside the squares just mentioned, precisely M further points of the graph.

We divide these M remaining points into sets, according to the values of their x-coordinates. Let the number of them which lie in the strip 0 < x ~ #1 be M 1 . And, for any i > 1, let the number which lie in the strip/*~-1 < x ~ #~ be M~. I f the number of/*'s is r, we obtain in this way a definite composition 4) of M,

M - = M l ~ - M s ~ . . . ~-M~ , ( 17 )

into r non-negative integers, this composition, like the partition (9), (10), being uniquely determined by the original partition (14) of N.

s) P.A. Macmahon, C o m b i n a t o r y Ana lys i s , II, 3. Our graph reads upwards, not downwards as in Macmahon.

4) A composition is a partition in which the order of the summands is important.

t 2 8

As a final consequence of (14), (15), we remark that (for each i = 1, 2 . . . . . r) the M t points of the i-th strip constitute, a translation apart, the graph of a certain partition P~ of M~, these partitions P~ being, just as much as the numbers M~ themselves, uniquely determined by (14). Further, for each i, the greatest part of P i is not greater t h a n / ~ . And, for each i > 1, the number of parts of P~ is not greater tha t #~-1 - - / ~ -

But, conversely, suppose that we choose any set of positive integers/~ satisfying (9) and (10), the sum of whose squares does not exceed N, and define M by (16); then choose any composition (17) of M, one par t M~ corresponding to each #~, taking care that

M, ~< Fz, ( p , _ l - ,u,) (i > 1) ;

and finally, for each Mi, choose arbitrarily a partition P~ having its greatest part not greater t h a n / ~ and having (for i > 1) not more than /~-i ~ ~ parts.

Then it is obvious that we can reverse our former construction at every step, and arrive at a definite partition (14) of N, which has n as its greatest part, and for which the corresponding #'s, M's and P ' s are preci- sely the ones we have chosen.

If, then, we denote by ~a,b(X) the generating function for the par- titions of N into at most a parts none of which exceed b, we have proved the identity

x~ _Z~,~l(x)~,_~,~,(x)~,_~,,~,(x)...x~:+~+..... (18) f,,(x) +)

the sum being taken over all partitions (9), (10) of n. But it is known 1) that, for finite a and b,

fa+b(X) ~l)a, b (X) = /a(Z ) lb (Z ) ,

while 1

1]) oo, b (X) : /b (X)

Substituting these values in (18), we obtain the required identity (13). This concludes the proof.

(Eingegangen den 17. September 1938.)

6) p . A. Maemahon, 1 o c. c i t . , 5.

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