Open access peer-reviewed chapter

# Group Theory from a Mathematical Viewpoint

Written By

Takao Satoh

Submitted: 07 July 2017 Reviewed: 31 October 2017 Published: 20 December 2017

DOI: 10.5772/intechopen.72131

From the Edited Volume

## Symmetry (Group Theory) and Mathematical Treatment in Chemistry

Edited by Takashiro Akitsu

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## Abstract

In this chapter, for the reader who does not major in mathematics but chemistry, we discuss general group theory from a mathematical viewpoint without proofs. The main purpose of the chapter is to reduce reader’s difficulties for the abstract group theory and to get used to dealing with it. In order to do this, we exposit definitions and theorems of the field without rigorous and difficult arguments on the one hand and give lots of basic and fundamental examples for easy to understand on the other hand. Our final goal is to obtain well understandings about conjugacy classes, irreducible representations, and characters of groups with easy examples of finite groups. In particular, we give several character tables of finite groups of small order, including cyclic groups, dihedral groups, symmetric groups, and their direct product groups. In Section 8, we deal with directed graphs and their automorphism groups. It seems that some of ideas and techniques in this section are useful to consider the symmetries of molecules in chemistry.

### Keywords

• group theory
• finite groups
• conjugacy classes
• representation theory
• character tables
• directed graphs
• automorphisms of graphs

## 1. Introduction

To make a long story short, a group is a set equipped with certain binary operation, for example, the set of all integers with the addition and the set of all n th power roots of unity with the multiplication. One of the origins of the group theory goes back to the study of the solvability of algebraic equations by Galois in the nineteenth century. He focused on the permutations of the solutions of an equation and gave rise to a concept of permutation groups. On the other hand, in 1872 Felix Klein proposed that every geometry is characterized by its underlying transformation groups. Here the transformation group means the group that comes from certain symmetries of the space. By using group theory, he classified Euclidean geometry and non-Euclidean geometry. As is shown earlier, groups have been established as important research objects on the study of permutations and symmetries of a given object. The group theory has achieved a good progress in modern mathematics and has various deep and sophisticated theories itself.

Today, the group theory has multiple facets and widespread applications in a broad range of science, including not only mathematics and physics but also chemistry. In chemistry, group theory is used to study the symmetries and the crystal structures of molecules. For each molecule, a certain group, which is called the point group, is defined by the symmetries on the molecule. The structure of this group reflects many physical and chemical properties, including the chirality and the spectroscopic property of the molecule. The group theory has become a standard and a powerful tool to study various properties of the molecule from a viewpoint of the molecular orbital theory, for example, the orbital hybridizations, the chemical bonding, the molecular vibration, and so on. In general, although each of modern mathematical theories is quite abstract and sophisticated to apply to the other sciences, the group theory has succeeded to achieve a good application by many authors, including Hans Bethe, Eugene Wigner, László Tisza, and Robert Mulliken. It seems that such expansions of mathematics to the other sciences are quite blessed facts for mathematicians.

Here we organize the contents of this chapter. First, we give mathematical notation and conventions which we use in this chapter. The reader is assumed to be familiar with elemental linear algebra and set theory. In Section 3, we review the definitions and some fundamental and important properties of groups. In particular, we show several methods to make new groups from known groups by considering subgroups and quotient groups. Then, we consider to classify known groups by using the concept of group isomorphism. In Section 4, we discuss and give many examples of finite groups, including symmetric groups, alternating groups, and dihedral groups. Then we give the classification theorem for finite abelian groups, which we can regard as an expansion of the Chinese remainder theorem. In Section 5, we consider to classify elements of groups by the conjugation and discuss the decomposition of a group into its conjugacy classes. In Section 6, we explain basic facts in representation theory of finite groups. In particular, we review representations of groups, irreducible representations, and characters. Finally, we give several examples of character tables of well-known finite groups. In Section 8, we consider finite-oriented graphs and their automorphisms. The automorphism group of a graph strongly reflects the symmetries of the graph. We remark that the reader can read this section without the knowledge of the facts in Sections 5 and 6.

## 2. Notation and conventions

In this section, we fix some notation and conventions and review some definitions in the set theory and the linear algebra:

Nthesetof natural numbers=123Zthesetof integers=0±1±2±3Qthesetof rational numbersRthesetof real numbersCthesetof complex numbers=a+b1abR
• For any a,bZ\0, the greatest common divisor of a and b is denoted by gcdab.

• For a set X, the cardinality of X is denoted by X. If X is a finite set, X means the number of elements of X.

• For sets X and Y, the difference of sets X and Y is denoted by X\YxxXxY.

• A map f:XY is surjective if for any yY; there exists some xX such that fx=y.

• A map f:XY is injective if fx=fx for x,xX; then x=x.

• A map f:XY is bijective if f is surjective and injective. In other words, the bijective map is one-to-one correspondence between X and Y.

• Let K be Q, R or C. For K-vector spaces V and W, a map f:VW is K-linear if f satisfies

fx+y=fx+fy,fkx=kfx

for any x,yV and kK.

• A linear map f:VV is called a linear transformation on V.

## 3. General group theory

In this section, we review elemental and fundamental topics in group theory, based on the authors’ book [1]. There are hundreds of textbooks for the group theory. Venture to say, if the reader wants to learn more from a viewpoint of symmetries, it seems to be better to see [2]. For high motivated readers, see [3, 4] for mathematical details.

### 3.1. Groups

Let G be a set. For any σ,τG, if there exists the unique element στG, which is called the product of σ and τ, such that the product satisfies the following three conditions, then the set G is called a group:

• (Associativity) For any σ,τ,ρG, στρ=στρ.

• (Unit) There exists some element eG such that for any σG,

eσ=σe=σ.

We call the element e the unit of G. According to the mathematical convention, we write 1G or simply 1, for the unit.

• (Inverse element) For any σG, there exists some element σG such that

σσ=σσ=e.

We call σ the inverse element of σ and write σ1.

If the definition of the product is clear from the content, we often omit the symbol and write στ instead of στ for simplicity. The product is a binary operator on G and is also called the multiplication of G.

Here we consider the following examples:

(E1) Each of the sets Z, Q, R, and C is a group with the usual addition. For the case Z, we see that the unit is 0 and for any nZ, the inverse of n is n. In general, if the product of a group G is additive, then G is called an additive group. We remark that N is not a group with the usual addition since any element does not have its inverse.

(E2) The set R×R\0 with the usual multiplication of real numbers forms a group. We see that the unit is 1 and for any rR×, the inverse of r is 1/r. We remark that R with the usual multiplication is not a group since 0 does not have its inverse. In general, if the product of a group G is multiplicative, then G is called a multiplicative group. Similarly, Q×Q\0 and C×C\0 are multiplicative groups.

(E3) For any nN n1, let Un be the set of n th power roots of unity:

Unexp21nC0kn1,

where

exp21ncos2n+1sin2n.

Then Un with the usual multiplication of C forms a group. Geometrically, Un is the set of vertices of the regular n-gon on the unit circle in the complex plane C. For example, U6 consists of the following points for ζ=exp2π16 in Figure 1.

In general, for a group G, if G consists of finitely many elements, then G is called a finite group. The number of elements of a finite group G is called the order of G, denoted by G. If G is not a finite group, then G is called an infinite group. The group Un is a finite group of order n, and the groups discussed in (E1) and (E2) are infinite groups.

(E4) Let K be Q, R, or C. We denote by M2K the set of 2×2 matrices with all entries in K:

M2KabcdabcdK.

Furthermore, we denote by GL2K the set of elements of M2K whose determinant is not equal to zero:

GL2KAM2KdetA0.

Then M2K with the usual addition of matrices forms an additive group. The unit of M2K is zero matrix, and for any A=aijM2K, its inverse is Aaij. Since GL2K does not have the zero matrix, the set GL2K is not an additive group. On the other hand, the set GL2K with the usual multiplication of matrices forms a multiplicative group. The unit of GL2K is the unit matrix E2, and for any A=aijGL2K, its inverse is the inverse matrix A1 as follows:

E21001,A1=1detAa22a12a21a11.

The group GL2K is called the general linear group of degree 2. Similarly, we can consider the general linear group GLnK of degree n for any nN.

Both M2K and GL2K are infinite groups. But the most significant difference between them is the commutativity of the products. Although we see A+B=B+A in M2K for any A,BM2K, the equation AB=BA does not hold in GL2K in general. For example, if A=1101 and B=1011, then we see

AB=2111,BA=1112.

For a group G, if στ=τσ holds for any σ,τG, then G is called an abelian group. The group GL2K is a non-abelian group, and all the groups as mentioned before except for GL2K are abelian groups.

### 3.2. Subgroups

Since group theory is an abstract itself, it had better for beginners to have sufficiently enough examples to understand it. In order to make further examples, we consider several methods to make new groups from known groups. The first one is a subgroup.

Let G be a group. If a nonempty subset H of G satisfies the following two conditions, then H is called a subgroup of G:

• For any σ,τH, στH.

• For any σH, σ1H.

We can consider H itself is a group by restricting the product of G to H. For any group G, the one point subset 1G is a subgroup of G. We call this subgroup the trivial subgroup of G. Let us consider some other examples:

(E5) The additive group Z is a subgroup of Q, R, and Z. For any nZ, the subset

nZ0±n±2nZ

of Z consisting of multiples of n is a subgroup of Z. Since 0Z=0 is the trivial subgroup, and since nZ=nZ, we usually consider the case nN.

(E6) Consider the group U6 consisting of 6th power roots of unity. Then we can consider U2 and U3 are subgroups of U6.

(E7) Let K be Q, R, or C. The subset

SL2KAGL2KdetA=1GL2K

of GL2K consisting of matrices whose determinants are equal to one is a subgroup of GL2K. We call SL2K the special linear group of degree 2.

In general, we can construct a subgroup from a subset of a group. Let S be a subset of a group G. Then the subset

Ss1e1s2e2smemmZ0siSei=±1

of G consisting of elements which are written as a product of some elements in S, and their inverses are a subgroup of G. Remark that if m=0, the product s1e1smem means 1G and that for any σ=s1e1s2e2smemS, its inverse is given by σ1=smemsm1em1s1e1. We call S the subgroup of G generated by S. The elements of S are called generators of the subgroup S. Here we give some examples:

(E8) The additive group Z is generated by 1. For any n1, the group Un of n th power roots of unity is generated by ζ=exp2π1/n. In general, a group generated by a single element is called a cyclic group. Thus, Z is an infinite cyclic group, and Un is a finite cyclic group. Remark that 1 and ζ1=exp2π1/n are also generators of Z and Un, respectively.

(E9) It is known that the additive groups Q, R, and C and the multiplicative groups GL2K and SL2K for K=Q,R,C are not finitely generated group.

Next, we consider a relation between the orders of a finite group and its subgroup. Let G be a group and H a subgroup of G. For any σG, the subset

σHστGτH

is called a left coset of H in G. We can see that σH=τH if and only if there exists some hH such that σ=τh.

(E10) In the additive group Z, for any nN, consider the subgroup nZ. Then, since the product of Z is written additively, a left coset of nZ is given by

σ+nZ=σ+τZ

for an element σZ. On the other hand, if we take the remainder r of the division of σ by n, then we see σ+nZ=r+nZ. Hence all left cosets of nZ in Z are given by

nZ,1+nZ,n1+nZ.

For simplicity, we write rn for r+nZ.

(E11) Consider the finite cyclic group U6 and its subgroup U2=±1 of order 2. Set ζexp2π1/6. Then we can see that

ζU2=±ζ=ζζ4=ζ4U2,ζ2U2=ζ5U2,ζ3U2=U2.

Hence there exist three left cosets of U2.

In example (E11), we can see that the order of U2 times the number of left cosets of U2 is equal to six, which is the order of U6. This is no coincidence. In general, for a finite group G and a subgroup H of G, the number of left cosets of H is called the index of H in G and is denoted by G:H. Then we have the following:

(Lagrange). As the above notation C, we have G=HG:H. Namely, the order of any subgroup of a finite group G is a divisor of G.

As a corollary, we obtain the following:

If G is a finite group of prime order, then G is a cyclic group.

### 3.3. Quotient groups

For a group G and its subgroup H, the set of left cosets of H is denoted by

G/HσHσG.

In general, this set does not have a natural group structure. Here we consider a condition to make it a group.

Let N be a subgroup of G. If σnσ1N for any nN and any σG, then N is called a normal subgroup of G. If G is abelian group, any subgroup of G is a normal subgroup. For a normal subgroup N of G, we define the product on G/N by using that on G. Namely, for any σN,τNG/N, define

σNτNστN.

Then this definition is well defined, and G/N with this product forms a group. The unit is 1GN=N, and for any σNG/N, its inverse is given by σN1=σ1N. We call G/N the quotient group of G by N.

(E12) The most important example for quotient groups is

Z/nZ=0n1nn1n

for nN. For any a,bZ, we have

an+bn=a+bn,an=an.

For example, in the group Z/6Z, we have

16+36=46,26+76=96=36,46=46=26.

For any 0rn1, since we see

rn=1n+1n++1nZ/nZ,

the group Z/nZ is a cyclic group of order n generated by 1n.

### 3.4. Homomorphisms and isomorphisms

As mentioned above, we have many examples of groups. Here, we consider relations between groups and examine which ones are essentially of the same type of groups. To say more technically, we classify groups by using isomorphisms.

Let G and H be groups. If a map f:GH satisfies

fστ=fσfτforanyσ,τG,

then f is called a homomorphism. A bijective homomorphism f:GH is called an isomorphism. Namely, an isomorphism is a map such that it is one-to-one correspondence between the groups and that it preserves the products of the groups. If G and H are isomorphic, we write GH.

(E13) Set

R>0xRx>0,

and consider it as a multiplicative subgroup of R×. The exponent map exp:RR>0 is an isomorphism from the additive group R to R>0.

(E14) Let K be Q, R, or C. Then the determinant map det GL2KK× is a homomorphism. It is, however, not an isomorphism since f is not injective. For example, det E2=detE2=1.

On the other hand, SL2K is a normal subgroup of GL2K. For any σ,τGL2K, we can see that

σSL2K=τSL2Kdetσ=detτ.

Define the map f:GL2K/SL2KK× by

σSL2Kdetσ.

Then f is an isomorphism. Indeed f is injective. For any xK×, if we consider the element σx001GL2K, we have fσSL2K=x. Hence f is surjective. Moreover, we have

fσSL2KτSL2K=fστSL2K=detστ=detσdetτ=fσSL2KfτSL2K.

(E15) For any nN, define the map f:Z/nZUn by knexp21/n. Then f is an isomorphism since f is bijective, and

fkn+ln=fk+ln=exp2k+lπ1/n=exp21/nexp21/n=fknfln.

Let G and H be isomorphic groups. Then, even if G and H are different as a set, they have the same structure as a group. This means that if one is abelian, finite or finitely generated, then so is the other, respectively. In other words, for example, an abelian group is never isomorphic to a non-abelian group and so on.

## 4. Finite groups

In this section, we give some examples of important finite groups.

### 4.1. Symmetric groups

For any nN, set X12n. A bijective map σ:XX is called a permutation on X. A permutation σ is denoted by

σ=12nσ1σ2σn.

Remark that this is not a matrix. We can omit a letter i 1in if the letter i is fixed. For example, for n=4:

12343241=134341

We call the permutation

ε12n12n

the identity permutation.

Let Sn be the set of permutations on X. For any σ,τSn, define the product of σ and τ to be the composition στ as a map. Then the set Sn with this product forms a group. We call it the symmetric group of degree n. The unit is the identity permutation, and for any σSn, its inverse is given by

σ1=σ1σ2σn12n.

The symmetric group Sn is a finite group of order n!.

Since S1 is the trivial group, and

S2=ε1221,

we see that Sn is abelian if n2. For n=3, we have

S3=ε122113312332123231123312,

and

12212332=123231123312=23321221.

Hence, S3 is non-abelian. Similarly, for any n3, Sn is non-abelian.

Here we consider another description of permutations. For distinct letters a1,,amX, the permutation

a1a2am1ama2a3ama1

is denoted by a1a2am and is called a cyclic permutation of length m. We call a cyclic permutation of length 2 a transposition. Namely, any transposition is of type

ij=ijji.

A cyclic permutation of length 1 is nothing but the identity permutation:

1=2==n=ε.

In general, a permutation cannot be written as a single cyclic permutation but a product of some cyclic permutations which do not have a common letter. For example, consider

σ=1234535412.

Then we see

σ:1341,252,

and hence

σ=13425=25134.

Remark that two cyclic permutations which do not have a common letter are commutative. For any cyclic permutation a1a2am, we have

a1a2am=a1a2a2a3am1am.

By using the above facts, we see

Every permutation can be written as a product of transpositions.

An expression of a permutation as a product of transpositions is not unique. For example,

132=1213=1323.

However, we have

For any permutation σ, consider expressions of σ as a product of transpositions. Then the parity of the number of transpositions is invariant.

For a permutation σ, if σ is written as a product of even (resp. odd) numbers of transpositions, then σ is called even permutation (resp. odd permutation). For example, the cyclic permutation a1a2am is even (resp. odd) permutation if m is odd (resp. even).

### 4.2. Alternating groups

In this subsection, we consider important normal subgroups of the symmetric groups. Let An be the set of even permutations of Sn. For any σAn, if we write σ as a product of transpositions, σ=τ1τk, then we see

σ1=τkτk1τ1An.

Clearly, if σ, τAn, then στAn. Thus, the subset An is a subgroup of Sn. We call An the alternating group of degree n. It is easily seen that An is a normal subgroup of Sn. For example, for n=3 and 4, we have

A3=ε123132,A4=ε123132124142134143234243123413241423.

For any σSn, we have

σAn=12An,ifσisoddpermutation,An,ifσis even permutation.

Hence Sn:An=2. Therefore, from Lagrange’s theorem, we see that An is a finite group of order n!/2.

### 4.3. Dihedral groups

For any nN n3, consider a regular polygon Vn with n sides, and fix it. A map σ:VnVn is called a congruent transformation on Vn if σ preserves the distance between any two points in Vn. Namely, σ is considered as a symmetry on Vn. Set

Dnσ:VnVnσisacongruent transformation.

For any σ,τDn, define the product of σ and τ to be the composition στ as a map. Then the set Dn with this product forms a group. We call it the dihedral group of degree n. The unit is the identity transformation.

Each congruent transformation on Vn is determined by the correspondence between vertices of Vn. Indeed, attach the number 1,2,,n to vertices of Vn in counterclockwise direction. For any σDn, if σ1=i, then the vertices 2,3,,n are mapped to i+1,i+2,,n,1,2,i1, respectively, Cor mapped to i1,i2,,1,n,n1,,i+1, respectively. If we express this by using the notation for permutations, we have

σ=12n1nii+1i2i1or12n1nii1i+2i+1.

The former case is a rotation, and the latter case is the composition of a rotation and a reflection. For n=3, see Figure 2. Thus the dihedral group Dn is a finite group of order 2n and is naturally considered as a subgroup of Sn. For n=3, since D3 is a subgroup of S3, and since both groups are of order 6, we see that D3=S3.

Let σDn be the rotation of Vn with angle 2πn in the counterclockwise direction and τDn be the reflection of Vn which fixes the vertex 1. Namely,

σ=12n1n23n1,τ=12n1n1n32.

Then the reflection of Vn which fixes the vertex i is written as σi1τσi1. Hence Dn is generated by σ and τ. Moreover, we have

Dn=1σσ2σn1τστσ2τσn1τ.

### 4.4. The structure theorem for finite abelian groups

In this subsection, we give a complete classification of finite abelian groups up to isomorphism. To begin with, we review the direct product of groups.

Let G and H be groups. Consider the direct product set

G×HghgGhH,

and define the product of elements gh,ghG×H to be

ghghgghh.

Then G×H with this product forms a group. The unit is 1G1H, and for any ghG×H, its inverse is given by g1h1G×H. We call the group G×H the direct product group of G and H. Similarly, for finitely many groups G1,G2,,Gn, we can define its direct product group G1××Gn. For each 1in, if Gi is a finite group of order mi, then G1××Gn is a finite group of order m1m2mn. The following theorem is famous in elementary number theory.

(Chinese remainder theorem). For any m,nN such that gcdmn=1. Then we have

Z/mnZZ/mZ×Z/nZ.

An isomorphism f:Z/mnZZ/mZ×Z/nZ is given by

xmnxmxn.

(E16) Consider the case m=2 and n=3. Each element x6 of Z/6Z is mapped to the following element by the above isomorphism f:

161213,262223=0223,363233=1203,46,4243=0213,565253=1223,060203.

(E17) If gcdmn1, the theorem does not hold. For example, consider the case of m=n=2. Any element xZ/2Z×Z/2Z satisfies that x+x is equal to zero. On the other hand, for the element y14Z/4Z, y+y is not equal to zero. Hence the group structures of Z/2Z×Z/2Z and Z/4Z are different.

Now, we show one of the most important theorems in finite group theory.

(structure theorem for finite abelian groups). Let G be a nontrivial finite abelian group. Then G is isomorphic to a direct product of finite cyclic groups of prime power order:

GZ/p1e1Z××Z/prerZ.

The tuple p1e1p2e2prer is uniquely determined by G, up to the order of the factors.

(E18) The list of finite abelian groups of order 72 up to isomorphism is given by

Z/9Z×Z/8Z,Z/9Z×Z/4Z×Z/2Z,Z/9Z×Z/2Z×Z/2Z×Z/2Z,Z/3Z×Z/3Z×Z/8Z,Z/3Z×Z/3Z×Z/4Z×Z/2Z,Z/3Z×Z/3Z×Z/2Z×Z/2Z×Z/2Z.

## 5. Conjugacy classes

In this section, we consider the classification of elements of a group by using the conjugation. The results of this section are used in Section 6.

Let G a group. For elements x,yG, if there exists some gG such that x=gyg1; then we say that x is conjugate to y and write xy. This is an equivalence relation on G. Namely, for any xG, we have xx by observing x=1Gx1G1. If xy, then x=gyg1 for some gG. Thus y=g1xg11, and hence yx. If xy and yz, then x=gyg1 and y=hzh1 for some g,hG. Thus x=ghzgh1, and hence xz. For any xG, the set

CxyGyx

is called the conjugacy class of x in G. If G is abelian group, for any xG, there exists no element conjugate to x except for x, and hence Cx=x. Here we give a few examples.

(E19) (Dihedral groups) For n3, the conjugacy classes of Dn are as follows:

1. If n is even:

1,σσ1,σ2σ2,,σn22σ2n2,σn2,τσ2τσn2τ,στσ3τσn1τ.

2.      If n is odd:

1,σσ1,σ2σ2,,σn12σ1n2,τστσn1τ.

Indeed, for the case where n is even, we can see the above from the following observation. For any xDn, since

xσix1=σjσiσj=σi,ifx=σj,σjτσiτσj=σi,ifx=σjτ,

the conjugates of σi are σ±i. On the other hand, for any xDn, since

xσiτx1=σjσiτσj=σi+2jτ,ifx=σj,σjτσiττσj=σi+2jiτ,ifx=σjτ,

the conjugates of σiτ are σkτ for any k such that kimod2. These facts induce Part (1).

(E20) (Symmetric groups) For any σSn, we can write σ as a product of cyclic permutations which do not have a common letter, like

σ=a1akb1blc1cm.

Furthermore, we may assume klm since the cyclic permutations appeared in the right hand side are commutative. Then we call klm is the cycle type of σ.

Elements σ,σSn are conjugate if and only if the cycle types of σ and σ are equal.

For example, conjugacy classes of S4 are given by

Cycle typeConjugacy class
11111S4
211121314232434
22123413241423
31{123,124,132,134, 142,143,234,243}
4{1234,1243,1324, 1342,1423,1432}

In the above examples, we verify that the number of elements of any conjugacy class is a divisor of the order of the group. In general, we have

Let G be a finite group. For any xG, Cx is a divisor of G.

## 6. Representation theory of finite groups

In this section, we give a brief introduction to representation theory of finite groups. There are also hundreds of textbooks for the representation theory. One of the most famous and standard textbooks is [5]. For high motivated readers, see [6, 7, 8] for mathematical details.

### 6.1. Representations

In this subsection, we assume that G is a finite group. Let V be a finite-dimensional C-vector space. Consider the following situation. For any σG and any vV, there exists a unique element σvV such that

1. σv+w=σv+σw,

2. σαv=ασv,

3. στv=στv,

4. 1Gv=v

for any σ,τG, αC and v,wV. Then we say that G acts on V and V is called a G-vector space.

The conditions (1) and (2) mean that for any σG, the map ρσ:VV defined by vσv is a linear transformation on V. Furthermore, from the conditions (3) and (4), we see that for any σG, the linear transformation ρσ1 is the inverse linear transformation of ρσ. Namely, each ρσ is a bijective. Set

GLVf:VVfisabijective linear transformation,

and consider it as a group with the product given by the composition of maps. Then we obtain the group homomorphism ρ:GGLV by σρσ. In general, for a finite group G and for a finite-dimensional C-vector space V, a homomorphism ρ:GGLV is called a representation of G. Then V is a G-vector space by the action of G on V given by

σvρσv

for any σG and vV. The dimension dimCV of V as a C-vector space is called the degree of the representation ρ. Observe the following examples:

(E21) For any finite group G, and any C-vector space V, we can consider the trivial action of G on V by σvv for any σG and vV. Namely, we can consider the homomorphism triv:GGLV by assigning σ to the identity map on V for any σG. This is called the trivial representation of G.

(E22) For any nN, consider the cyclic group Un and the action of Un on C given by the usual multiplication exp21/nzexp21/nz of the complex numbers for any kZ and zC. The action of exp21/n on C is the rotation on C in the counterclockwise direction centered at the origin with angle 2/n. If we take 1C as a basis of the C-vector space C, we can identify GLC with the general linear group GL1C=C× by considering the matrix representation. Under this identification, the corresponding representation ρ:UnGLC=C× is given by the natural inclusion map Un´C×.

(E23) Consider the symmetric group S3 and the numerical vector space C3. The group S3 naturally acts on C3 by the permutation of the components given by

σx1x2x3xσ11xσ12xσ13.

If we take the standard basis e1,e2,e3 as a basis of C3, we can identify GLC3 with the general linear group GL3C by considering the matrix representation. Under this identification, the corresponding representation ρ:S3GLC3=GL3C is given by σeσ1eσ2eσ3. Similarly, we can obtain the representation ρ:SnGLCn=GLnC that is given by

σeσ1eσ2eσn.

This is called the permutation representation of Sn.

Next we consider subrepresentations of a representation. Let ρ:GGLV a representation. If there exists a subspace W of V such that

σwWρσwW

for any σG and wW, then W is called a G-subspace of V. For any σG, the restriction ρσW:WW of ρσ is a bijective linear transformation on W, and we obtain the representation ρW:GGLW given by σρσW. It is called a subrepresentation of ρ.

(E24) Consider the permutation representation ρ:S3GLC3=GL3C as in (E23). Let us consider subspaces

W1xxxxC,W2xyzxyzCx+y+z=0

of C3. It is easily seen that these are S3-subspaces and the subrepresentation ρW1 is the trivial representation. Geometrically, W1 and W2 in C3 are drawn in Figure 3. In a precise sense, if we naturally consider R3 as a subset of C3, then Figure 3 shows W1R3 and W2R3 in R3.

For a G-vector space V, if there exist G-subspaces W1 and W2 of V such that any element vV can be uniquely written as

v=w1+w2w1W1w2W2,

then V is called the direct sum of W1 and W2 and is written as V=W1W2. Similarly, we can define the direct sum of G-subspaces W1,W2,,Wm for any m3. Let ρ, ρW1, and ρW2 be the correspondent representations of G to V, W1, and W2, respectively. We also say that the representation ρ is the direct sum of ρW1 and ρW2.

(E25) As the notation in (E24), V is the direct sum of W1 and W2. Indeed, for the standard basis e1,e2,e3 of V, we see that e1+e2+e3 and e1e2,e1e3 are bases of W1 and W2, respectively. Thus, for any x=x1e1+x2e2+x3e3C3, we can rewrite

x=x1+x2+x33e1+e2+e3+x12x2+x33e1e2+x1+x22x33e1e3.

Furthermore, we verify that this expression is unique by direct calculations.

In general, we have

(Maschke). Let ρ:GGLV a representation and W a G-subspace of V. Then there exists a G-subspace W such that V=WW.

### 6.2. Irreducible representations

In subsection 4.4, we have discussed the classification of finite abelian groups by using the concept of group isomorphisms. Here we consider the classification of finite-dimensional representations of finite groups by using irreducible representations and equivalence relations among representations.

Let G be a finite group and ρ:GGLV its representation. The trivial subspaces 0 and V are G-subspaces of V. If V has no G subspace other than these, V is called the irreducible G-space, and ρ is called the irreducible representation of G.

(E26) Any one-dimensional representation is trivial. For example, the representation ρ:UnGLC=C× in (E23) is irreducible. Let us consider the other example. For any σSn, set

sgnσ1ifσis even permutation,1ifσisoddpermutation.

Then we can easily see that the map sgn:SnC×=GLC is a homomorphism and, hence, is a representation of Sn. This irreducible representation is called the signature representation of Sn.

(E27) As the notation in (E24), ρW1 is irreducible since it is one-dimensional. The representation ρW2 is also irreducible. Indeed, if W2 is not irreducible, there exists a one-dimensional G-subspace W in W2 since W2 is a 2-dimensional G-vector space. Take wW (w0). Then w is an eigenvector of ρW2σ for any σS3. However, we can see that there is no such vector in W2 by direct calculations.

By observing (E25), (E26), and (E27), we see that C3 is a direct sum of the irreducible G-subspaces W1 and W2. In general, by using Maschke’s theorem above, we obtain.

For any representation ρ:GGLV of a finite group G, the G-vector space V can be written as a direct sum of some irreducible G-subspaces. Namely, ρ can be written as sum of some irreducible representations of G.

Remark that the expression of a direct sum of irreducible representations is not unique in general. For example, let ρ:GGLC2 be the trivial representation. Then for the standard basis e1,e2 of C2, we have

C2=Ce1Ce2=Ce1Ce1+e2=Ce1Ce1+2e2=.

In order to do the classification of representations, we consider the equivalency of representations. Let ρ1:GGLV1 and ρ2:GGLV2 be representations of G. If there exists a bijective linear map ι:V1V2 such that

ισv=σιv,σG,vV1,

then we say that V1 is isomorphic to V2 as a G-vector space and write V1V2. We also say that ρ1 is equivalent to ρ2 and write ρ1ρ2.

(E28) For any group G. let unit:GGLC=C× be the trivial representation of G. Then any trivial representation ρ:GGLV is equivalent to unit. The representation unit is called the unit representation of G.

The following theorem is one of the most important theorems in representation theory of finite groups.

Let G be a finite group.

1. The number of irreducible representations of G up to equivalent is finite. Furthermore, it is equal to the number of the conjugacy classes of G.

2. For any representation ρ:GGLV, ρ is equivalent to a direct sum of some irreducible representations:

VW1W2Wm.

Furthermore, the tuple of the components is uniquely determined by G, up to the order.

### 6.3. Characters

In this subsection, for a given representation, we give a method to determine whether it is irreducible or not by using characters. Let ρ:GGLV be a representation. Take a basis v1,,vn of V, and fix it. By using this basis, we can consider ρσ as an n×n-matrix Aσ=aij, which is the matrix representation of ρσ. Then set

χρσTrAσ=a11+a22++annC

for any σG. Remark that this definition is well defined since it does not depend on the choice of a basis of V. Indeed, if w1,,wn is another basis of V, the matrix representation of ρσ with respect to this basis is given by P1AσP for a some regular matrix P. Hence TrP1AσP=TrAσ. We call the map χρ:GC the character of ρ. Remark that for elements σ,τG, if στ, then ρσρτ in GLV. Thus, χρσ=χρτ. Namely, χρ is constant on each of the conjugacy classes of G.

(E29) Consider the example (E25). Let ρ:S3GLC3 be the permutation representation of S3. The conjugacy classes of S3 are as follows:

Cycle typeConjugacy class
1111S3
21121323
3123132

Hence, in order to calculate the values of the character χρ of ρ, it suffices to calculate its values on 1S3, 12, and 123. If we take the standard basis e1,e2,e3 of C3, we have ρσ=eσ1eσ2eσ3, and hence

χρ1S3=3,χρ12=1,χρ123=0.

In general, as in (E29), for a representation ρ:GGLV, χρ1G is the degree of the representation, which is equal to dimCV.

Now, we define the inner product of characters. For complex functions φ,ψ:GC on G, set

φψ=1GσGφσψσ¯

where z¯ means the complex conjugation of zC. We call it the inner product of ϕ and ψ. The following theorems are quite important and useful from the viewpoint to find and to calculate all of the irreducible representations.

1. (Orthogonality) Let ρi:GGLVi (i=1,2) be irreducible representations. Then

χρ1χρ2=1ifρ1ρ2,0ifρ1/ρ2.

2. For a representation ρ:GGLV,

ρisirreducibleχρχρ=1.

(E30) We have the three irreducible representations of S3. By direct calculations, we obtain the following list:

σ1S3121323123132
χunitσ111111
χsgnσ111111
χρW2σ200011

Hence we see that in each of cases, we have χρχρ=1.

By Theorem 6.3, we see that for any representation ρ:GGLV, V can be written as

VW1m1W2m2Wkmk

where each Wi is an irreducible G-vector space and Wi is not isomorphic to Wj as a G-vector space if ij. For each 1ik, the number mi is called the multiplicity of Wi in V.

As the notation above, let ρi be the irreducible representation of G correspond to the G-vector space Wi. Then we have

1. χρ=m1χρ1++mkχρk.

2. χρχρi=mi.

Namely, each of the multiplicity of the irreducible G-vector spaces in V is calculated by the inner product of the characters

3.     G=i=1kχρi12.

Namely, the sum of the squares of the degrees of the irreducible representations is equal to the order of G.

From the above theorems, we verify that if we want to know all irreducible representations of G, it suffices to calculate its characters. The list of all values of all characters is called the character table of G. Finally, we give a few examples of the character tables of finite groups.

(E31) Observe (E30). Since we have

χunit12+χsgn12+χρW212=4+1+1=6=S3,

it turns out that unit, sgn, and ρW2 are all irreducible representations of S3 up to equivalence. Hence the list in (E30) is the character table of S3.

(E32) Consider the cyclic group Un. Since Un is abelian, any conjugacy class consists of a single element, and there exist n conjugacy classes. Hence there exist n distinct irreducible representations. Now, for any 0ln1, define the map ρl:UnGLC=C× by

ζkζkl0kn1

where ζ=exp2π1/n. Then we obtain

σ1Unζζ2ζn1
χρ0σ11111
χρ1σ1ζζ2ζn1
χρn1σ1ζn1ζn2ζ

Hence we see that ρ0,ρ1,,ρn1 are nonequivalent one-dimensional representations, and hence the above list is the character table of Un. In general, all irreducible representations of an abelian group are of degree 1.

(E33) (Dihedral groups) For n3, consider the dihedral groups Dn. First, for any a,b=±1, there exist the four one-dimensional representations εa,b:DnC× defined by

εa,bx=1akifx=σk,1ak+bifx=σkτ.

These maps are characterized by the images of σ and τ, which are 1a and 1b, respectively. Next, for any 1ln1, we can consider the two-dimensional representations ρl:DnGL2C given by

ρlx=cos2klπ/nsin2klπ/nsin2klπ/ncos2klπ/nifx=σk,cos2klπ/nsin2klπ/nsin2klπ/ncos2klπ/n0110ifx=σkτ.
1. The case where n is even. For any 1ln22, since we can see χρlχρl=1 by direct calculation, ρl s are irreducible representations of Dn. Since we have

χε1,112+χε1,112+χε1,112+χε1,112+χρ112++χρn2212=2n=Dn,

it turns out that εa,b and ρl for a,b=±1 and 1ln22 are all irreducible representations of Dn up to equivalence. The character table of D4 is give as follows:

x1D4{σ,σ3σ2στσ3ττσ2τ
χε1,1x11111
χε1,1x11111
χε1,1x11111
χε1,1x11111
χρ1σ20200

ii. The case where n is odd. Similarly, we can see that ε1,b and ρl for b=±1 and 1ln12 are all irreducible representations of Dn up to equivalence. The character table of D5 is give as follows:

x1D5σσ4σ2σ3τστσ4τ
χε1,1x1111
χε1,1x1111
χρ1σ22cos2π/52cos4π/50
χρ2σ22cos4π/52cos2π/50

## 7. Direct products

In chemistry, groups appear in symmetries of molecules. The structures of some of them are given by direct products of finite groups. Here we consider direct product groups and its irreducible representations.

Let G and H be finite groups. Set

G×HghgGhH,

and define the product on G×H by

ghghgghh.

Then G×H with this product forms a group. This is called the direct product group of G and H. The unit is 1G1H, and the inverse of gh is g1h1. If G and H are finite groups, then it is clear that G×H=GH. For conjugacy classes C and C of G and H, respectively, the direct product set C×C is a conjugacy class of G×H, and any conjugacy class of G×H is obtained by this way.

In order to construct irreducible representations of G×H, we consider tensor products of vector spaces. For G-vector space V and H-vector space W, let F be the vector space with basis vwvVwW and R the subspace of F generated by

v1+v2wv1wv2w,vw1+w2vw1vw2,αvwαvw,vαwαvw,

for any v,v1,v2V, w,w1,w2W, and αC. The quotient vector space F/R is called the tensor product of V and W and is denoted by VW. The coset class of vw is denoted by vw. If v1,,vm and w1,,wn are bases of V and W, respectively, then elements viwj (1im and 1jn) form a basis of VW. Hence dimVW=dimVdimW.

For any gG and hH, we can define the action of G×H on VW by

ghi=1mj=1nαijviwji=1mj=1nαijgvihwj,

and hence, VW is a G×H-vector space. For the representations ρ:GGLV and ρ:GGLW corresponding to the G-vector spaces V and W, respectively, we denote by ρρ:GGLVW the representation corresponding to the G×H-vector space VW. Then we have

(1) As the notation above, if ρ and ρ are irreducible, so is ρρ.

(2) If ρ1,,ρk (resp. ρ1,,ρl) are all irreducible representations of G (resp. H) up to equivalence, then ρiρj (1im and 1jn) are all irreducible representations of G×H up to equivalence.

(E34) For V=C and W=C, the tensor product VW of V and W is a one-dimensional C-vector space with basis 11. Thus, we have a bijective linear map VWC given by

a11a.

In general, we identify CC with C through this map.

Let us consider the direct product U2×U3. Under the identification CC=C, the character table is given as follows:

σ111ζ1ζ2111ζ1ζ2
χρ0ρ0σ111111
χρ0ρ1σ1ζζ21ζζ2
χρ0ρ2σ1ζ2ζ1ζ2ζ
χρ1ρ0σ111111
χρ1ρ1σ1ζζ21ζζ2
χρ1ρ2σ1ζ2ζ1ζ2ζ

where ζ=exp2π1/3.

(E35) Consider the direct product U2×S3. Its character table is given as follows:

σ11S31ij1ijk11S31ij1ijk
χρ0unitσ111111
χρ0sgnσ111111
χρ0ρW2σ201201
χρ1unitσ111111
χρ1sgnσ111111
χρ1ρW2σ201201

## 8. Graphs and their automorphisms

In this section, we consider directed graphs and their automorphism groups. Here we do not assume for the reader to know the facts in Sections 5 and 6.

### 8.1. Graphs

According to literatures, there are several different definitions of a graph. Briefly Ca directed graph Γ consists of vertices and oriented edges whose endpoints are vertices. (For details for the definition of graphs, see page 14 of [9].) For an oriented edge e, we denote by ie and te the initial vertex and the terminal vertex of e. Each oriented edge e has the inverse edge e¯ such that e¯e and e¯¯=e. It is clear that ie¯=te and te¯=ie. An oriented edge e such that ie=te is called a loop. For any v,wVΓ, we assume that there may exist more than one oriented edge whose initial vertex is v and terminal vertex w. If this is the case, we say that Γ has multiple oriented edges.

(E36) A directed graph is easy to understand if it is drawn by a picture. See Figure 4. The vertices v,w,x,y,z are depicted by small circles. The oriented edges a,b,c,d,e,f,g,h are depicted by arrows from the initial vertex to the terminal vertex, and their inverse edges are omitted for simplicity.

We denote by VΓ and EΓ the sets of the vertices and the oriented edges of Γ, respectively. If both VΓ and EΓ are finite set, we call Γ a finite graph. Here, we consider only finite graphs. Remark that EΓ is always even since EΓ is written as e1e¯1eme¯m. For any v,wVΓ, if there exists a successive sequence of oriented edges such that the initial vertex of the first edge is v and the terminal vertex of the last edge w, then the graph is called a connected graph. For example, see Figure 5. In the following, we assume that all graphs are connected.

### 8.2. Automorphisms of graphs

Let Γ and Γ be graphs. A morphism of directed graphs from Γ to Γ is a map

σ:VΓEΓVΓEΓ

which maps vertices to vertices and edges to edges, such that

σie=iσe,σte=tσe,σe¯=σe¯

for any eEΓ. Namely, σ maps the initial vertex, the terminal vertex, and the inverse edge of an oriented edge to those of the corresponding oriented edge, respectively. For simplicity, we write σ:ΓΓ. If σ is bijective, then it is called an isomorphism. An isomorphism from Γ to Γ is called an automorphism of Γ. Let AutΓ be the set of all automorphisms of Γ. Then AutΓ with the composition of maps forms a group. We call it the automorphism group of Γ. Let us consider a few easy examples of AutΓ.

(E37) See Figure 6. The graph Γ1 consists of one vertex v and two oriented edges e and e¯. Hence all morphisms from Γ1 to Γ1 are automorphisms since if σ:ΓΓ is a morphism, then σv=v, and σe=e or σe=e¯. If σe=e, then σe¯=e¯ as a consequence, and hence σ is the identity map on Γ. If σe=e¯, then σe¯=e as a consequence, and hence σ is the orientation-reversing automorphism on Γ. Thus, AutΓ1=σ1σ2Z/2Z where σ1e=e and σ2e=e¯.

On the other hand, the graph Γ2 consists of one vertex v and four oriented edges e, e¯, f, and f¯. It is easily seen that there are eight possible automorphisms on Γ2. Namely, all of them map v to v, and the correspondences of edges are given by

σ1:efef,σ2:efe¯f,σ3:efef¯,σ4:efe¯f¯,σ5:effe,σ6:eff¯e,σ7:effe¯,σ8:eff¯e¯.

Hence AutΓ2=σ1σ8. It turns out that σ2, σ3, and σ5 are generators of AutΓ2. In (E41), we study the structure of AutΓ2 more.

Next, in order to describe the group structure of AutΓ more simply, we consider semidirect products of groups. For high motivated readers, see [10] for details and more examples. The semidirect product groups are kinds of generalizations of direct product groups. Let G be a group, K a subgroup of G, and H a normal subgroup of G. Furthermore, if we have

G=hkhHkK,HK=1G,

then we call G the semidirect product group of H and K and denote it by G=HK.

(E38) Recall the dihedral group Dn=1σσ2σn1τστσ2τσn1τ. Set H1σσ2σn1 and K1τ. Then we can see that the subset H is a normal subgroup of Dn, HK=1, and Dn=hkhHkK. Thus Dn=HK.

Remark that for any gG, we can write g=hk for some hH and kK and that this expression is unique. Namely, if g=hk=hk for h,hH and k,kK, then we have h1h=kk1HK. Hence h1h=kk1=1G, and hence h=h and k=k. Therefore, if G<, we see that G=HK. We also remark that if hk=kh for any hH and kK, then G is isomorphic to the direct product group of H and K, namely, GH×K. Thus, the semidirect product is a generalization of the direct product.

Now, let Γ be a graph. For any v,wVΓ, we number the oriented edges of Γ with v as initial vertex and w as terminal vertex. Then every oriented edge e can be uniquely represented as e=vwk. In particular, we can arrange the numbering such that e¯=wvk for any e=vwkEΓ.

(E39) See Figure 7. We can arrange a numbering of the oriented edges as

e=vw1,e¯=wv1,f=vw2,f¯=wv2,g=vw3,g¯=wv3,h=ww1,h¯=ww2.

Let T be the subgroup of AutΓ consisting of automorphisms that fix all vertices pointwise:

TtAutΓtv=vvVΓ.

Let M be the subgroup of AutΓ consisting of automorphisms that fix the numberings of edges:

MmAutΓmvwk=vwkforanyvwVandanynumberk.

Then we have AutΓ=TM

(E40) Recall the graph Γ1 in (E37). Since every automorphism fixes the vertex v, we see that AutΓ1=T and M=1. Similarly, if a graph Γ has only one vertex, then AutΓ=T.

(E41) Recall the graph Γ2 in (E37). We have AutΓ2=T and M=1. Set Hσ2σ3 and Kσ5. Then it is seen that HZ/2Z×Z/2Z, KZ/2Z, and AutΓ2HK.

(E42) Consider the directed graph Γ depicted as the regular n-gon. Then we see that T=1 since if an automorphism fixes all vertices then it must fix all edges. Thus, AutΓ=M. Furthermore, we can see that MDn=στ where σ is the 2π/n-angled rotation and τ is the reflection.

(E43) Consider the directed graph Γ in Figure 8. We arrange a numbering of the oriented edges as

e=wv1,e¯=vw1,f=wv2,f¯=vw2,g=wv3,g¯=vw3.

The subgroup T consists of automorphisms which permute the oriented edges e,f,g, and hence TS3. On the other hand, the subgroup Q consists of two automorphisms given by the identity map and

σ:vwwv,efge¯f¯g¯,

and hence QZ/2Z. Therefore AutΓS3Z/2Z.

The readers are strongly encouraged to consider further examples by oneself. It makes their understandings better and deeper.

As a remark, we mention the irreducible representations of a semidirect product group. As mentioned in Section 7, the irreducible representations of a direct product group G×H can be calculated with those of G and H. The situation for semidirect products groups, however, is much more complicated. In general, in order to study the irreducible representations of semidirect product groups, we require some arguments in advanced algebra.

## Acknowledgments

The author would like to thank Professor Takashiro Akitsu, who is a chemist of our faculty, for introducing to him this work and many useful comments. He considers it a privilege since this is the first interaction across disciplines as a mathematician. He also would like to thank Professor Naoko Kunugi, who is a mathematician majoring in the representation theory of finite groups, for her useful comments about references of the field.

A part of this work was done when the author stayed at the University of Bonn in 2017. He would like to express his sincere gratitude to the Mathematical Institute of the University of Bonn for its hospitality and to Tokyo University of Science for its financial supports.

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Written By

Takao Satoh

Submitted: 07 July 2017 Reviewed: 31 October 2017 Published: 20 December 2017