Contents

1 INTRODUCTION 3

2 GROUP FUNDAMENTALS 52.1 Groups and Subgroups . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.4 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.5 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.6 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.7 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.8 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.9 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.10 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.11 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Permutation Groups . . . . . . . . . . . . . . . . . . . . . . . 82.2.1 definition . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Cayley’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.1 Theorem (Cayley) . . . . . . . . . . . . . . . . . . . . . 9

3 GROUP ACTION ON A SET 103.1 Group Action . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1.3 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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3.1.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.8 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 ISOTROPY SUBGROUPS 174.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 174.1.2 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 184.1.3 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 ORBITS 195.1 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.2.1 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6 APPLICATIONS OF G-SETS TO COUNTING 236.1 Theorem (Burnside’s Formula ) . . . . . . . . . . . . . . . . . 25

6.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 276.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 286.1.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 286.1.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 296.1.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7 APPLICATION PROBLEMS RELATED G-SET AND G-SET ON COUNTING 317.1 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.2 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

8 CONCLUSION 38

9 REFERENCES 40

2

Chapter 1

INTRODUCTION

In algebra and geometry a group action is a way of describ-ing symmetries of objects using groups.The essential elementsof the objects are described by the symmetry group of this set,which consists of bijective transformation of set.In this case,thegroup is also called a permutation group (especially if set is fi-nite or not a vector space) or transformation groups.A groupactions is an extention to the definition of a symmetry group inwhich every element of the group ”acts” like a bijective trans-formation (or”symmetry”) of the set ,without being identifiedwith that transformation.This allows for more comprehensivedescription of the symmetries of an object,such as a polyhedronby allowing the same group to act on several different sets offeatures such as the set of vetices ,the set of edges and the setof faces of the polyhedron.

Historically,the first group action studied was the action ofthe Galois group on roots of polynomial.However ,there arenumerous examples and applications of group action in manybranches of mathematics ,including algebra,topology geometry,number theory and analysis as well as the science including

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chemistry and physics.

The abstraction provided by Group action is powerful one,because allows geometrical idea to the applied to more abstractobject .Many objects in mathematics have natural group actiondefined on them .In particular ,group can act on other groups ,oreven on themselves .Despite this generality the theory of groupaction contains wide reaching theorems ,which can to prove deepresult in several fields.And the set of group action on a set whatare the changes in the set.

Here the outline of sections .In section 1 we will discuss Fun-damentals of Groups and definition of Sub groups ,section 2 dis-cuss the Group Action on Set ,section 3 describes the isotropysubgroups with some examples,section 4 describes Orbits andan important theorem,section 5 describes Applications of GroupAction on a Set to Counting including the formula and some ex-amples section 6 describes the application problems of relatedGroup Action and it’s application which may help you to seethe real beauty of group action on a set.

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Chapter 2

GROUP FUNDAMENTALS

This chapter define some fundamental idea about groups andsubgroups and also giving some examples of them

2.1 Groups and Subgroups

2.1.1 Definition

Let S be a non-empty set .Any function f :→ S × S is calledbinaray operation on S or equivalently ,an operation (?) on S

is called a binary operation on S (or S is closed under theoperation (?)).if a ? b ∈ S ,for all a, b ∈ S

2.1.2 Example

The equation (+), (·)and (−) all are binary operations on Z,Q,Rand C,where Z is the set of all integers ,Q is the set of all ratio-nal numbers,R is the set of all real numbers,and C is set of allcomplex numbers

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2.1.3 Definition

A non empty set S with one or more than one binary operationsis called a mathematical system or(algebraic system) and if (?)isa binary operation on S,then we say that (S, ?) is a mathemat-ical system

2.1.4 Definition

An element e of a mathematical system (?) is called an identity-element of S if a ? e = a = e ? a, for all a ∈ S

2.1.5 Definition

Let(S, ?),be a mathematical system with an identity element e.An element a ∈ S is said to have an inverse in S if there existsb ∈ S such that a ? b = e = b ? a(Note that ,b is called inverse ofa and is denoted by a−1 and we write a−1 = b)

2.1.6 Definition

A Group is a non empty setG on which there is defined a binaryoperation (a, b)→ ab satisfying the following propertiesclosure :If a and b belongs to G, then ab is also in GAssociativity : a(bc)=(ab)c ,for all a,b,c ∈ G;Identity :There is an element 1∈ G such that a1=1a=a for alla ∈ G;Inverse :if a is in G,then there is an element aa−1 = a−1a = 1

A group G is abelian if the binary operation is commutativei.e, ab=ba for all a,b in G. In this case binary operation is oftenwritten additively ((a, b) −→ a+ b),with the identity as 0 rather

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than 1.There are some very familiar examples of abelian groups un-der addition,namely the integers Z,rationals Q,the real numbersR,the complex numbers C,and the integers Zm modulo m

2.1.7 Definition

A subgroup H is a non empty subset of G that forms a group un-der the binary operations of G ,Equivalently ,H is a non emptysubset of G such that if a and b belongs to H.so does ab−1.If A is any subset of group G,the subgroup generated by A is thesmallest subgroup containing A,denoted by 〈A〉.It is obvious that 〈e〉 and G are always subgroup of group G

2.1.8 Definition

Let (G, .)and(G′, ∗) be groups.A map φ : G → G′, such thatφ(x.y) = φ(x) ∗ φ(y),for all x, y ∈ G is called homomorphism

2.1.9 Definition

The map φ : G → G′ is called an isomorphism and G and G′

are said to be isomorphic if

φ is a homomorphismφ is a bijection

2.1.10 Definition

A group G is Cyclic if G can be generated by a simple element,i.e, there is some element aεG such that G = {an : n ∈ G}(here

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the operation is multiplication).A finite cyclic group generatedby ′a′ is necessarily abelian

2.1.11 Definition

The order of an element a in G (|a|)is the least positive integern such that an = 1;if no such integer exists ,the order of X isinfinite .Thus |a| = n ,then the cyclic subgroup〈a〉 generated bya has exactly n elements ,and ak = 1 if and only if k is a multipleof n.The order of the group G ,denoted by |G|, is simply the numberof elements in G

2.2 Permutation Groups

2.2.1 definition

A permutation of a set S is a bijection on S,that is, a functionπ : S → S that is one− to− one and on− to.(If S is finite theπ is one-to-one iff it is on-to)

2.2.2 Definition

There are several permutation groups that are of major inter-ests.the set Sn of all the permutations of {1, 2, 3, .....n} is calleda symmetric group on n letters.The subgroup An of all even per-mutations {1, 2, 3, .....n} is called the aleternatimg groups on n letters

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2.3 Cayley’s Theorem

2.3.1 Theorem (Cayley)

Every finite group G can be embedded in a symmetric group.Proof

To each g ∈ G ,define the left multiplication function Ig :G→ G,where Ig(x) = gx for each x ∈ G.Each Ig is a permuta-tion of G as a set with inverse Ig−1.So Ig belongs to Sym(G) .Since Ig1◦Ig2 = Ig1g2 (i.e, g1(g2(x))=(g1g2)x for all x ∈ G) associ-ating g to Ig give a homomorphism of groups ,G→ Sym(G).Thishomomorphism is one to one since Ig determine g (after allIg(e) = g) . Therefore the corresponding g → Ig is an em-bedding of G as a subgroup of Sym(G).

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Chapter 3

GROUP ACTION ON A SET

3.1 Group Action

Definition 2.1.1 Defines a binary operation ∗ on a set S to be afunction mapping S × S into S.The function ∗ gives us a rulefor ”multiplying” an element s1inS and an element s2 to yieldan element s1 ∗ s2 inS.More generally ,for any sets A,B,and C,we can view a map,∗ : A × B → C as defining multiplication ,where any elementaonA times any element b of B has as value some element c ofC .Of course ,we write a ∗ b = c,or simply ab = c.In this section,we will be concerned with the case where X us a set , G is agroup, and we have a map∗ : G×X → X.we shall write ∗(g, x)as g ∗ x or gx

3.1.1 Definition

Let X be a set and G a group.An action of G on X is a map∗ : G×X → X such that1. ex = x for all x ∈ X,2. (g1g2)(x) = g1(g2x) for all x ∈ and all g1, g2 ∈ G

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Under conditions ,X is a G-set.

3.1.2 Example

Let X be any set , and let H be a subgroup of the group Sx ofall permutations of X.Then X is an H-Set, where the action ofσ ∈ H on X is its action as an element of Sx , so thatσx = σ(x)for all x ∈ X. condition 2 is a consequence of definition of per-mutation multiplication as function composition ,and condition1 is immediately from the definition of the identity permutationas an identity function ,Note that ,in particular,{1, 2, 3, .....n} inan Snset

Our next theorem will show that for every G-set X and eachg ∈ G, the map σg : X → X defined by σg(x) = gx is a per-mutation of X, that is a homomorphism ∅ : G→ SX such thatthe action of G on X is essentially the example above action ofimage subgroup H = ∅[G]ofSX on X .so actions of subgroups ofSX on X describe all possible group actions on X.when studyingthe set X ,actions using subgroups of SX suffice.However ,some-times a set X is used to study G via a group of G on X.Thus weneed the more general concept given by the definition.

3.1.3 Theorem

Let X be a G-set for each g ∈ G,then function σg : X → Xdefined by σg(x) = gx for x ∈ X is a permutations of X .Also,the map ∅ : G → Sx defined by ∅(g) = σg is a homomorphismwith the property that ∅(g)(x) = gx.

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proof

To show that σg is permutation of X ,we must show that itis a one to one map of X onto itself.suppose that σg(x1) = σg(x2)s for x1, x2 ∈ X.then gx1 = gx2consequently, g−1(gx1) = g−1(gx2)

using condition 2 in definition ,we see that(g−1g)x1 = (g−1g)x2,so ex1 = ex2.condition 1 of the definition then yieldsx1 = x2so σg is a one-to-one.the two conditions of the definition shows that for x ∈ X

we have σg(g−1x) = g(g−1)x = (gg−1x)S = ex = x

so σg maps X onto X ,thus σg is indeed a permutation,To show that ∅ : G → SX defined by ∅(g) = σg is a homomor-phism,we must show that

∅(g1g2) = ∅(g1)∅(g2) for all g1g2 ∈ Gwe show the equality of these two permutations in SX by show-ing they both carry an x ∈ X into same element.using the twoconditions in definition and rule for function composition,Weobtain

∅(g1g2)(x) = σg1g2(x)= (g1g2)x

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= g1(g2x)= g1σg2(x)= σg1(σg2(x))= (σg1σg2)(x)

= (∅(g1)∅(g2))(x)

Thus ∅ is a homomorphism.The stated property ∅ follows atonce since by our definitions ,we have ∅(g)(x) = σg(x) = gx.

It follows from the preceding theorem that if X G−set, then thesubset of G leaving every element of X fixed is normal subgroupNofG , and we can regard X as a G/N -set where the action of acoset gN on X is given by(gN)x = gx for each x ∈ X,N = {e},then the identity element of G is the only element that leavesevery x ∈ X fixed ; we say that G acts faithfully on X.A groupG is transitive on a G−set Xif for each x1, x2 ∈ X, there existsg ∈ G such that gx1 = gx2.Note that G is transitive on X if andonly if the subgroup ∅[G] ofSX is transitive on X , as defined incoming examples .

we continue with more examples G− sets.

3.1.4 Example

Every group G is itself is a G− set , where the action on g2 ∈ Gbyg1 ∈ G is given by left multiplication .That is ∗(g1, g2) = g1g2.IfH is a subgroup ofG , we can also regardG as anH−set,where∗(h, g) = hg.

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3.1.5 Example

Let H be a subgroup of G .Then G in an H − set under an con-jugation where ∗(h, g) = hgh−1 for g ∈ G and h ∈ H .condition1 is obvious , and condition 2 note that

∗(h1h2, g) = (h1h2)g(h1h2)−1 = h1(h2gh

−12 )h−11 = ∗(h1, ∗(h2, g))

We always write this action ofHonG by conjugation as hgh−1.Theabbreviation hg described before the definition would cause ter-rible confusion with group operation of G.

3.1.6 Example

For students who have studied vector spaces with real (or com-plex) scalars , We mention that the axioms (rs)V = r(sV)and 1V=V for scalars r and s and a vector V showsthat the set of vectors is an R?− set or(C?− set) for multiplica-tive group of nonzero scalars.

3.1.7 Example

Let H be a subgroup of G , and let LH be the be the set of allleft cosets of H .Then LH is G− set ,where the action of g ∈ Gon the left coset xH is given by g(xH) = (gx)H.Observe thatthis action is well defined :if yH = xH,then y = xh for someh ∈ H, and

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g(yH) = (gy)H = (gxh)H = (gx)(hH) = (gx)H = g(xH).

A series of exercises shows that every G − set is isomorphicto one that may be formed using these left coset G − set asbuilding blocks.

3.1.8 Example

Let G be the group D4 ={ρ0, ρ1, ρ3, ρ3, µ1, µ2, δ1, δ2} of symme-tries of squares .In figure show the square with vertices 1,2,3,4as in a square .we label the sides s1, s2, s3, s4, the diagonalsd1 and d2 , vertical and horizontal axes m1,m2,the centerC,and mid point Pi of sides si.Recall that ρi corresponds torotating the square counterclockwise through πi/2 radians ,µicorresponds to flipping on the axis mi , and δi to flipping on thediagonal di.we let

X = {1, 2, 3, 4, s1, s2, s3, s4,m1,m2, d1, d2, C, P1, P2, P3, P4}

Then X can be regarded as a D4 − set in a natural way .Ta-ble describes completely the complete action of D4 on X and isgiven to provide geometric illustrations of ideas to be introduced.We should be sure that we understand how this table is formedbefore continuing.

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Figure 3.1: FIGURE

Figure 3.2: TABLE

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Chapter 4

ISOTROPY SUBGROUPS

4.1 Definition

Let X be a G− set .Let x ∈ X and g ∈ G. It will be importantto know when gx = x

We let

Xg = {x ∈ X|gx = x}and GX = {g ∈ G|gx = x}

4.1.1 Example

For the D4 setX in the example in the previous chapter,we have,

Xρ0 = XXρ1 = {C}

Xµ1= {s1, s3,m1,m2, C, P1, P3}

Also , with G = D4,

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G1 = {ρ0, δ2}Gs3 = {ρ0, µ1}

Gd1 = {ρ0, ρ2, δ1, δ2}We have the computation of the other Xσ GX to in the prob-lem section.

Note that the subsets GX given in the preceding example were,ineach case , subgroups of G.This is true in general.

4.1.2 Theorem

Let X be a G−set . Then GX is a subgroup of G for each x ∈ X

proof

Let x ∈ X and let g1, g2 ∈ GX .

Then g1x = x and g2x = x.consequently, (g1g2)x = g1(g2x) = g1x = x, so g1g2 ∈ GX , GX isclosed umder the induced operation of G .Ofcourse ex = x, so

e ∈ GX .

If g ∈ GX ,then gx = x,so x = ex = (g−1g)x = g−1(gx) =g−1x,and consequently g−1 ∈ GX is a subgroup of G.Thus GX is subgroup of G

4.1.3 Definition

Let X be a G − set and let x ∈ X.The subgroup GX is theisotropy subgroup of x

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Chapter 5

ORBITS

For the D4 − set X in the example of group action on a setwith the action table ,the elements in the subset {1, 2, 3, 4} arecarried into elements of this same subset under action by D4

.Further more ,each element 1,2,3 and 4 is carried in to all theother elements of the subset by the various elements of D4 .Weproceed to show that every G − set X can be partitioned intosubsets of this types

5.1 Theorem

Let X be a G − set .For x1, x2 ∈ X,let x1 ∼ x2 ∈ X,letx1 ∼ x2 ∈ X if and only if there exists g ∈ G such that gx1 = x2.Then ∼ is an equivalence relation on X.

proofFor each x ∈ X ,we have ex = x, so x ∼ x and ∼ is reflexive.suppose x1 ∼ x2, so gx1 = x2 for some g ∈ G.Then g−1x2 = g−1(gx1) = (g−1g)x1 = ex1 = x1,so x1 ∼ x2 and’∼’ is symmetric.Finally ,if x1 ∼ x2 and x2 ∼ x3, then g1x1 = x2 and g2x2 = x3

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for some g1g2 ∈ G .Then (g2g1)x1 = g2(g1x1) = g2x2 = x3,so x1 ∼ x3 and ∼ is tran-sitive

5.2 Definition

Let X be a G−set .Each cell in the partition of equivalence rela-tion described in the above theorem is an orbit in X under Gif x ∈ X ,the cell containing x is the orbit of x .we let this cellbe GX .

The relationship between the orbits in X and the group struc-ture of G lies at the heart applications that appear in comingchapter .The following theorem gives this relationship .Recallthat for a set X,we use |X| for the number of elements , and(G : H) is the index of a subgroup H in group G.

5.2.1 Theorem

Let X be a G − set and let x ∈ X.Then |GX | = (G : GX). If|G| is finite then |GX | is a divisor of |G|

proofWe define a one-to-one map ψ from GX onto the collection ofleft cosets of GX in G .let x1 ∈ GX .Then there exists a g1 ∈ Gsuch that g1x = x1.We define ψ(x1) to be the left coset g1GX .We must show that

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this map is well defined,independent of the choice of g1 ∈ G suchthat g1x = x1.Suppose also that g′1x = x1.then g1x = g′1x,so g−11 (g1x) = g−11 (g′1x),from which we deducex = (g−11 g′1)x.therefore g−11 g′1 ∈ GX ,so g′1 ∈ g1GX ,and g1GX = g′1GX .Thus wemap ψ is well defined.

To show that the map ψis one to one ,suppose x1, x2 ∈ GX ,and ψ(x1) = ψ(x2).Then there exists g1, g2 ∈ G.Such that x1 = g1x, x2 = g2x, and g2 ∈ g1GX .Theng2 = g1g forsome g ∈ GX ,

so x2 = g2x = g1(gx) = g1x = x1.Thus ψis one -to-one

Finally ,we show that each left cosets of GX in G is of theform ψ(x1),for some x1 ∈ GX .Let g1GX be a left coset .Then ifg1x = x1 ,we gave g1GX = ψ(x1).Thus ψ maps GX one to oneonto collection of right cosets so |GX | = (G : GX).

If |G| is finite ,then the equation |G| = |GX |(G : GX) showsthat |GX | = (G : GX) is a divisor of |G|.

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5.2.2 Example

LetX be the D4− set in the example given in the chapter groupaction on a set with action table given bye Tables.With G = D4

,we have G1 = {1, 2, 3, 4} and G1 = {ρ0, δ2}. Since |G| =8, wehave|G1| = (G : G1)=4

We should remember not only the cardinality equation in theabove theorem but also that the elements of G carrying x intog1x are precisely the elements of the left coset g1GX .Namely,ifg ∈ GX ,then (g1g)x = g1(gx) = g1x.On the other hand ,if g2x =g1x, then g−11 (g2x) = x so (g−11 g2) ∈ GX so g2 ∈ g1 GX .

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Chapter 6

APPLICATIONS OF G-SETSTO COUNTING

This section present an application of our work with G − set

to counting.suppose for example, ,we wish to count how manydistinguishable ways the six faces of a cube can be marked withfrom one to six dots to form a die .The standard die is markedso that when placed on a table with 1 on the bottom and the2 toward front ,the 6 is on top ,the 3 on the left,4 on the right,and the 5 on the back . of course , other ways of marking thecube to give a distinguishably different die are possible.

Let us distinguish between the faces of the faces of the cubefor the moment and call them the bottom,top,left ,right,frontand back.Then the bottom can have any one of six marks fromone dot to six dots ,the top any one six marks from one dots tosix dots ,the top any one of the five remaining marks , and soon .there are 6!=720 ways the cube faces can be marked in all.Some markings yield the same die as others ,in some sense thatone marking can be carried into another by a rotation of the

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marked cube.For example if the standard die is described aboveis90◦ counterclockwise as we look down on it, then 3 will be onthe front face rather than 2,but it is the same die.

There are 24 possible position of cube on a table ,for any oneof six faces can be placed down , and then any one of four tothe front,giving 6.4=24 possible positions .any positions can beachieved from any other by rotation of the die .these rotationform a group G ,which is isomorphic to a subgroup of S8 .Welet X be the 720 possible ways of marking the cubes and letG acts on X by rotation of the cube.We consider two markingto give the same die if one can be carried into the other underaction by an element of G ,that is ,by rotating the cube .In theother words , we consider each orbit in X under G to correspondto single die and ,different orbits to give different dice .The de-termination of the number of distinguishable dice thus leads tothe question of determining the number of orbits under G in aG− set X

The following theorem gives a tool for determining the num-ber of orbits in a G − set X under G.Recall that for eachg ∈ G we let Xg be the set of elements of X left fixed by g ,sothatXg = {x ∈ X|gx = x}.Recall also that for each x ∈ X,welet GX = {g ∈ G|gx = x} , and GX is the orbit of x under G.s

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6.1 Theorem (Burnside’s Formula )

Let G be a finite group and X a finite G-set .if r is the numberof orbits in X under G ,then

r.|G| =∑g∈G|Xg| (6.1)

proofWe consider all pairs (g, x) where gx = x , and let N be thenumber of such pairs.For each g ∈ G, there are |Xg| pairs hav-ing g as first member .Thus,

N =∑g∈G|Xg| (6.2)

On the other hand ,for each x ∈ X there are |Gx| pairs havingx as second member .thus also we have

N =∑x∈X|Gx| (6.3)

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Bye the theorem 5.2.1 we have |Gx| = (G : Gx). But we knowthat(G : Gx) = |G|/|Gx|, So we obtain |Gx| = |G/|Gx|.Then

N =∑x∈X

|G||Gx|

= |G|(∑x∈X

1

|Gx|) (6.4)

Now 1/|Gx has the same value for all x in the same orbit ,andif we let % be any orbit, then

∑x∈%

1

|Gx|=

∑x∈%

1

|%|= 1 (6.5)

substituting (6.5)in (6.4),we obtain

N = |G|(number of orbits in X under G) = |G|.r(6.6)

Comparison of Eq 6.3 and 6.6 gives Eq6.1

corollary

If G is finite group and X is a finite group G− set, then

(number of orbits in X under G =1

|G|.∑g∈G|Xg|

(6.7)proof

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The proof of this corollary follows immediately from the pre-ceding theorem

Let us continue our computation of the number of distinguish-able dice as our first example

6.1.1 Example

we let X be the set of 720 different markings of faces of a cubeusing from one to six dots.Let G be the group of 24 rotationof the cube as discussed above.We saw that the number of dis-tinguishable dice is the number of orbits in X under G . now|G|=24 . For g ∈ G where g 6= e,we have |Xg|=0.because anyrotation other than the identity element charges anyone of the720 markings into a one.However |Xe| =720 ,since the identityelement leaves all 720 marking fixed .Then by corollary

(number of orbits )= 124 .720 = 30

so there are 30 dice.

Of course the number of dice could be counted without usingthe machinery of the preceding corollary ,but by using elemen-tary combinatorics as often taught in a freshman finite mathcourse.In marking a cube to make a die,we can, by rotationif necessary, assume the face marked as 1 is down .There arefive choices for the top (opposite) face.By rotating the die aswe look on it , any one of the remaining four faces could bebrought to the front position ,so there are no different choicesinvolved for the front face.But with respect to the number on

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the front face there are 3.2.1 possibilities for the remaining threeside faces.Thus there are 5.3.2.1=30 possibilities in all.

6.1.2 Example

How many distinguishable ways we can seven people be seatedat a round table where there is no distinguishable ”head” to thetable?

Of course 7! possible arguments .A rotation of people achievedby asking each person to move one place to right results in thesame arrangement.Such a rotation generates a cyclic group G oforder 7,which consider to act on X in the obvious way .Again,only the identity e leaves any arrangement fixed and it leavesall 7!.arrangement fixed.

s(number of orbits)=17 .7!

=17 .7.6!=6!

=720

6.1.3 Example

How many distinguishable necklace (with no clasp)can be madeusing seven differtent colored beads of the same size ?Unlikethe above example the necklace can be turned over as well asrotated.Thus we consider the full dihedral group D7 of order2.7=14 as acting on the set X of 7! as acting on the set X of 7!possibilities .Then the number of distinguishable necklace is

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(number of orbits)= 114 .7! = 360

in using the corollary we compute |G| and |Xg| for each g ∈ Gwill pose no real problem .Let us give an example where |Xg| isnot as a trivial to compute as in the preceding example,We willcontinue to assume knowledge of very elementary combinatorics

6.1.4 Example

Let us find the number of distinguishable ways the edges of anequilateral triangle can be painted if four different colors of paintare available ,assuming only one color is used on each edge ,andthe same color ,may be used on different edges.

Of course there are 43 ways of painting edges in all ,sinceeach of the three edges may be any one of four colors .We canconsider X to be the set of these 64 possible painted triangle thegroup G acting on X is the group of symmertries of the trianglewhich is isomorphic to S3 we use the notation for elements in S3

Which we consider to be S3.we use the notation fo the elementsinS3.We need to compute |Xg|for each of the six elements g in S3

|Xρ0| = 64 Every painted triangle is left fixed ρ0|Xρ1| = 4 To be invariant under ρ1, all edges must be samecolor and there 4 possible colors|Xρ2| = 4 same reason for ρ1|Xµ1| = 16 the edges that are interchanges must be the same

color (4 possibilities )and other edges also be any of the color

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(time 4 possibilities)|Xµ2| = |Xµ1

| = 16 same reason as for µ1

Then

∑g∈S3

|Xg| = 64 + 4 + 4 + 16 + 16 + 16 = 120 (6.8)

Thus (number of orbits)=16 .120 = 20

and there are 20 distinguishable painted triangle

6.1.5 Example

We repeat the penultimate example with the assumption that adifferent color is used on each edge.The number of each possibleways of painting the edges is then 4.3.2=24 and let X be theset of 24 possible painted triangles again the group acting on Xcan be consider to be S3.Since all edges are different color wesee |Xρ0|=24 while |Xg|=0 forg 6= ρ0 thus

(number of orbits )=16 .24=4

so there are 4 distinguishable triangles.

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Chapter 7

APPLICATION PROBLEMSRELATED G-SET ANDG-SET ON COUNTING

7.1 Computation

1. In this letX = {1, 2, 3, 4, s1, S2, S3, S4,m1,m2d1, d2, C, P1, P2, P3, P4}be the D4 set of example inthe sectiongroup action on setwith the action table .find the following were G = D4

a) The fixed set Xσ for each σ ∈ D4, i.e,Xρ0, Xρ1, Xρ2, ........Xδ2

b) The isotropy subgroupGxfor each X ∈ X that isG1, G2, G3.......GP3

, GP4

a)the orbits in X under D4

Answer

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a)Xρ0 = XXρ1 = {C}Xρ2 = {µ1, µ2, d1, d2, C}Xρ3 = {C}Xµ1

= {ρ1, ρ2,m1,m2, C, P1, P3}Xµ2

= {ρ2, ρ4,m1,m2, C, P2, P4}Xδ1 = {2, 4, d1, d2, C}Xδ2 = {1, 3d1, d2, C}b)G1 = G3 = {ρ0, δ2}G2 = G4 = {ρ0, δ1}Gs1 = Gs3 = {ρ0, µ1}Gs2 = Gs4 = {ρ0, µ2}Gm1

= Gm2= {ρ0, ρ2, µ1, µ2}

Gd1 = Gd2 = {ρ0, ρ2, δ1, δ2}GC = G,GP1

= GP3= {ρ0, µ1}, GP2

= GP4= {ρ0, µ2}

G1 = G2 = G3 = G4 = {1, 2, 3, 4}, Gs1 = Gs2 = Gs3 =Gs41 = {s1, s2, s3, s3}, Gµ1

= Gµ2= {m1,m2}, Gd1 = Gd2 =

{d1, d2}, GC = {C}, GP1= GP2

=, GP3= GP4

={P1, P2, P3, P4}

2. Find the number of orbits in {1, 2, 3, 4, 5, 6, 7, 8} under thecyclic subgroup < (1, 3, 5, 6) > of S8.

Answer

Let X = {1, 2, 3, 4, 5, 6, 7, 8}We have | X1 |= 8| X(1,3,5,6) |=| {2, 4, 7, 8} |= 4 | X(1,6,5,3) |=| {2, 4, 7, 8} |= 4

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Therefore we have

∑g∈G| Xg |= 8 + 4 + 4 + 4 = 20 (7.1)

Therefore (no.of orbits )=14 .(20) = 5

3. Find the number of orbits in {1, 2, 3, 4, 5, 6, 7, 8} under thesubgroup of S8 generated by (1,3)and (2,4,7)AnswerThe group G generated by (1,3) and (2,4,7) has order 6.Let X = 1, 2, 3, 4, 5, 6, 7, 8

| X1 |= 8| X1,3 |= 6| X2,4,7 |= 5| X2,7,4 |= 5| X(1,3)(2,4,7) |= 3| X1,3)(2,4,7) |= 3

Then∑g∈G| Xg |= 8 + 6 + 5 + 5 + 3 + 3 = 30 (7.2)

(no.of orbits)=16 .30 = 5

4. Find the number of distinguishable tetrahedral dice thatcan be made using one,two,three,and four dots on the faceof regular tetrahedron ,rather than a cube?AnswerThe group of rigid motion of the tetrahedron has 12 ele-ments because anyone of four triangle can be on the bottomand tetrahedron can be then be rotated through 3 positions

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,keeping the sane face on the bottom.We see that | Xg |= 0unless g is the identity ’i’ of this group ’G’,and | X4 |=4!=24

Thus there are 112(24) = 2,distinguishable tetrahedral dice.

5. A wooden cubes of the same size are to be painted on differ-ent color on each face to make children’s blocks .How manydistinguishable blocks can be made eight colors of paint areavailable?Answerthe total number of ways such a block can be painted withdifferent colors on each face is 8.7.6.5.4.3 the group of rigidmotions of the cube has 24 elements .The only rigid mo-tion leaving unchanged a block with different colors on allfaces the identity which leaves all such blocks fixed .Thusthe number of distinguishable block is 1

24 .(8.7.6.5.4.3) =8.7.5.3 = 40.2! = 840

7.2 Concepts

1. Let X and Y be G − set with the same group G .An iso-morphism between G− set X and Y is a map

φ : X → Y ,that is one -one Y satisfies g(φ(x))=φ(gx) forall x ∈ X and g ∈ G The two G−set are isomorphic.If suchan isomorphism between them exists,Let x be D4 − set

(a) Find two distinct orbits of X that are isomorphicsubset D4-sets.(b) Show that the orbits {1, 2, 3, 4} and {S1, S2, S3, S4}are not isomorphic subD4sets

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[Hint:Find an element of G that essentiallydifferent fashion on the two orbits]

(c) Are the orbits gave your answer to part (a) the onlytwo different isomorphic sub-D4set of X?

Answer(a) {P1, P2, P3, P4} and {S1, S2, S3, S4} are isomprphic Sub-D4-sets.Note that if you change each P to an S in actiontable.We get a duplication of four columns for S1, S2, S3, S4

(b) δ1 leaves two elements 2 and 4 of {1, 2, 3, 4} fixed butδ1 leaves no elements of S1, S2, S3, S4 fixed(c) Yes for the part(b) the only other conveciable choice foran isomorphism is {m1,m2} with {d1, d2}.However µ1 leavesthe element of {m1,m2} fixed and moves both elements of{d1, d2}.So they are not isomorphic.

2. Let X be the D4 set(a) Does D4 act faithfully on X(b) Find all orbits in X on which D4 act faithfully as asub-D4-setAnswer(a) Yes ,because identity is the only element that leaves allelements fixed .Hence for ρ0 is the only element of G thatleaves every element of X fixed .(b) {1, 2, 3, 4},{S1, S2, S3, S4},{P1, P2, P3, P4

7.3 Theory

1. Let X be a G-set and Y⊆X .Let Gy : {g ∈ G|gy = y for all y ∈ Y }show that Gy is a subgroup of G

35

AnswerGiven that X is a G-set ,i.e G has an action on x ∀ x ∈ X.G has the same actionon Y.Therefore Y⊆XLet y ∈ Y and g1, g2 ∈ Gy

g1y = yg2y = y

(g1g2)y = g1(g2y) =g1y =y , ∀y ∈ Y

Therefore g1g2 ∈ Gy

Therefore Gy is closed under the operation we have e ∈ Ysince ey = y , ∀ y ∈ YLet g ∈ Gy,then

gy = y

g−1(gy) = g−1y

(g−1g)y = g−1y

ey = g−1yy = g−1y ∀ y ∈ Yy−1 ∈ Gy

Therefore Gy ⊆ G

2. Let G be the additive group of real numbers .Let the actionθ ∈ G on the real plane R2 be given by rotating the planecounter clockwise about the origin through θ radians .LetP be a point other than the origin in the plane(a) Show that R2 is G-s(b) Describe geometrically the orbit containing p(c) Find the group GP

Answer

36

(a) Because rotation through 0 radians leaves each pointof the plane fixed.The first requirement of the definitionsatisfied ,the second requirement (θ1 + θ2)P = θ1(θ2)P isalso valid ,because a rotation counterclockwise through θ2radians and then θ1 radians .(b) The orbit P is circle with center at the origin (0,0) andthe radius the distance from P to the origin .(c) The group GP is the cyclic subgroup < 2φ > of of G

3. Let {Xi|i ∈ I} be a disjoint collection of sets ,so Xi∩Xj = φ

for i 6= j .Let each Xi be a G-set for the same group G(a) Show that ∪i∈IXi can be viewed in a natural way as aG-set ,in the union of the G-set X(b) Show that every G-set X is the union of the orbitsAnswer(a)

Let X = ∪i∈IXi and let x ∈ X

Then x ∈ Xi foe precisely one index .i ∈ I because the aredisjoint and we define gx for each g ∈ G to be value given bythe action of G of Xi ,conditions (1)and(2) in the definition3.1.1 are satisfied because Xi is a G-set by assumption .(b) We have see that each orbit in X is a sub-G-set .ThenG-set X can be regarded as the union of these sub-G-setbecause the action gx of g ∈ G on x ∈ X coincides withthe sub-G-Set action gx of g ∈ G on the same element xviewed as an element of it’s orbit

37

Chapter 8

CONCLUSION

In this project we have done a deep study of group-action ona set with many examples and it’s application in G-set countingand also some problem relating .

From the study of group-action onset we reach the follow-ing conclusions :one way of thinking of G acting on X is thatelements of group .G may be applied to elements of X to givea new elements of X .And the goal to build automorphism ofsets(permutation) by using groups.Additionally we have stud-ied the influences of G-set on counting.Group action on a setis an advanced topic in abstract algebra. Group action on aset and it’s application give another mode of finding solution tiproblems,mainly in coloring problems,it is more understandablethan the other methods.And in daily life we use group action ona set without knowing ourselves.

Through this project we have understand what do we mean bygroup action on a set , isotropy of group and orbits with someexamples, and also we have understood the application of G-seton counting with examples and some problems.Over all through

38

this project we get an idea about group action on a set with it’sapplication on counting.

39

Chapter 9

REFERENCES

John B.Fraleigh -A First Course In Abstract Algebra(7th

Edition)-Pearson Education India (2008)I.N Herstein - Topics in Algebra (2th Edition)-John Wiley andsons (2006)M.R Adhikari - Groups,Rings and Modules WithApplications-Universities press (India)Private Limited (2003)P.B.Bhattacharaya,S.K.Jain,S.R.Nagapaul - Basic AbstractAlgebra(2th Edition)-Cambridge university press (1995)en.Wikipedia.org/wiki/groupen.wikipedia.org/wiki/groupactionwww.Groupaction.com

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