Open access peer-reviewed chapter

Group Theory from a Mathematical Viewpoint

By Takao Satoh

Submitted: July 7th 2017Reviewed: October 31st 2017Published: December 20th 2017

DOI: 10.5772/intechopen.72131

Downloaded: 291

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 nth 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 aand bis denoted by gcdab.

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

  • For sets Xand Y, the difference of sets Xand Yis denoted by X\YxxXxY.

  • A map f:XYis surjective if for any yY; there exists some xXsuch that fx=y.

  • A map f:XYis injective if fx=fxfor x,xX; then x=x.

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

  • Let Kbe Q, Ror C. For K-vector spaces Vand W, a map f:VWis K-linear if fsatisfies

fx+y=fx+fy,fkx=kfx

for any x,yVand kK.

  • A linear map f:VVis 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 Gbe a set. For any σ,τG, if there exists the unique element στG, which is called the product of σand τ, such that the product satisfies the following three conditions, then the set Gis called a group:

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

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

eσ=σe=σ.

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

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

σσ=σσ=e.

We call σthe inverse element of σand write σ1.

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

Here we consider the following examples:

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

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

(E3) For any nNn1, let Unbe the set of nth power roots of unity:

Unexp21nC0kn1,

where

exp21ncos2n+1sin2n.

Then Unwith the usual multiplication of Cforms a group. Geometrically, Unis the set of vertices of the regular n-gon on the unit circle in the complex plane C. For example, U6consists of the following points for ζ=exp2π16in Figure 1.

Figure 1.

The sixth roots of unity.

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

(E4) Let Kbe Q, R, or C. We denote by M2Kthe set of 2×2matrices with all entries in K:

M2KabcdabcdK.

Furthermore, we denote by GL2Kthe set of elements of M2Kwhose determinant is not equal to zero:

GL2KAM2KdetA0.

Then M2Kwith the usual addition of matrices forms an additive group. The unit of M2Kis zero matrix, and for any A=aijM2K, its inverse is Aaij. Since GL2Kdoes not have the zero matrix, the set GL2Kis not an additive group. On the other hand, the set GL2Kwith the usual multiplication of matrices forms a multiplicative group. The unit of GL2Kis the unit matrix E2, and for any A=aijGL2K, its inverse is the inverse matrix A1as follows:

E21001,A1=1detAa22a12a21a11.

The group GL2Kis called the general linear group of degree 2. Similarly, we can consider the general linear group GLnKof degree nfor any nN.

Both M2Kand GL2Kare infinite groups. But the most significant difference between them is the commutativity of the products. Although we see A+B=B+Ain M2Kfor any A,BM2K, the equation AB=BAdoes not hold in GL2Kin general. For example, if A=1101and B=1011, then we see

AB=2111,BA=1112.

For a group G, if στ=τσholds for any σ,τG, then Gis called an abelian group. The group GL2Kis a non-abelian group, and all the groups as mentioned before except for GL2Kare 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 Gbe a group. If a nonempty subset Hof Gsatisfies the following two conditions, then His called a subgroup of G:

  • For any σ,τH, στH.

  • For any σH, σ1H.

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

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

nZ0±n±2nZ

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

(E6) Consider the group U6consisting of 6th power roots of unity. Then we can consider U2and U3are subgroups of U6.

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

SL2KAGL2KdetA=1GL2K

of GL2Kconsisting of matrices whose determinants are equal to one is a subgroup of GL2K. We call SL2Kthe special linear group of degree 2.

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

Ss1e1s2e2smemmZ0siSei=±1

of Gconsisting 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 s1e1smemmeans 1Gand that for any σ=s1e1s2e2smemS, its inverse is given by σ1=smemsm1em1s1e1. We call Sthe subgroup of Ggenerated by S. The elements of Sare called generators of the subgroup S. Here we give some examples:

(E8) The additive group Zis generated by 1. For any n1, the group Unof nth 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, Zis an infinite cyclic group, and Unis a finite cyclic group. Remark that 1and ζ1=exp2π1/nare also generators of Zand Un, respectively.

(E9) It is known that the additive groups Q, R, and Cand the multiplicative groups GL2Kand SL2Kfor K=Q,R,Care not finitely generated group.

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

σHστGτH

is called a left coset of Hin G. We can see that σH=τHif and only if there exists some hHsuch that σ=τh.

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

σ+nZ=σ+τZ

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

nZ,1+nZ,n1+nZ.

For simplicity, we write rnfor r+nZ.

(E11) Consider the finite cyclic group U6and its subgroup U2=±1of 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 U2times the number of left cosets of U2is equal to six, which is the order of U6. This is no coincidence. In general, for a finite group Gand a subgroup Hof G, the number of left cosets of His called the index of Hin Gand 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 Gis a divisor of G.

As a corollary, we obtain the following:

If Gis a finite group of prime order, then Gis a cyclic group.

3.3. Quotient groups

For a group Gand its subgroup H, the set of left cosets of His 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 Nbe a subgroup of G. If σnσ1Nfor any nNand any σG, then Nis called a normal subgroup of G. If Gis abelian group, any subgroup of Gis a normal subgroup. For a normal subgroup Nof G, we define the product on G/Nby using that on G. Namely, for any σN,τNG/N, define

σNτNστN.

Then this definition is well defined, and G/Nwith 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/Nthe quotient group of Gby 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/nZis a cyclic group of order ngenerated 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 Gand Hbe groups. If a map f:GHsatisfies

fστ=fσfτforanyσ,τG,

then fis called a homomorphism. A bijective homomorphism f:GHis 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 Gand Hare isomorphic, we write GH.

(E13) Set

R>0xRx>0,

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

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

On the other hand, SL2Kis 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 fis an isomorphism. Indeed fis injective. For any xK×, if we consider the element σx001GL2K, we have fσSL2K=x. Hence fis surjective. Moreover, we have

fσSL2KτSL2K=fστSL2K=detστ=detσdetτ=fσSL2KfτSL2K.

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

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

Let Gand Hbe isomorphic groups. Then, even if Gand Hare 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:XXis called a permutation on X. A permutation σis denoted by

σ=12nσ1σ2σn.

Remark that this is not a matrix. We can omit a letter i1inif the letter iis fixed. For example, for n=4:

12343241=134341

We call the permutation

ε12n12n

the identity permutation.

Let Snbe the set of permutations on X. For any σ,τSn, define the product of σand τto be the composition στas a map. Then the set Snwith 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 Snis a finite group of order n!.

Since S1is the trivial group, and

S2=ε1221,

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

S3=ε122113312332123231123312,

and

12212332=123231123312=23321221.

Hence, S3is non-abelian. Similarly, for any n3, Snis non-abelian.

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

a1a2am1ama2a3ama1

is denoted by a1a2amand is called a cyclic permutation of length m. We call a cyclic permutation of length 2a transposition. Namely, any transposition is of type

ij=ijji.

A cyclic permutation of length 1is 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 σas a product of transpositions. Then the parity of the number of transpositions is invariant.

For a permutation σ, if σis written as a product of even (resp. odd) numbers of transpositions, then σis called even permutation (resp. odd permutation). For example, the cyclic permutation a1a2amis even (resp. odd) permutation if mis odd (resp. even).

4.2. Alternating groups

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

σ1=τkτk1τ1An.

Clearly, if σ, τAn, then στAn. Thus, the subset Anis a subgroup of Sn. We call Anthe alternating group of degree n. It is easily seen that Anis a normal subgroup of Sn. For example, for n=3and 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 Anis a finite group of order n!/2.

4.3. Dihedral groups

For any nNn3, consider a regular polygon Vnwith nsides, and fix it. A map σ:VnVnis called a congruent transformation on Vnif σpreserves the distance between any two points in Vn. Namely, σis considered as a symmetry on Vn. Set

Dnσ:VnVnσisacongruent transformation.

For any σ,τDn, define the product of σand τto be the composition στas a map. Then the set Dnwith this product forms a group. We call it the dihedral group of degree n. The unit is the identity transformation.

Each congruent transformation on Vnis determined by the correspondence between vertices of Vn. Indeed, attach the number 1,2,,nto vertices of Vnin counterclockwise direction. For any σDn, if σ1=i, then the vertices 2,3,,nare 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 Dnis a finite group of order 2nand is naturally considered as a subgroup of Sn. For n=3, since D3is a subgroup of S3, and since both groups are of order 6, we see that D3=S3.

Figure 2.

The transformations of the regular triangle.

Let σDnbe the rotation of Vnwith angle 2πnin the counterclockwise direction and τDnbe the reflection of Vnwhich fixes the vertex 1. Namely,

σ=12n1n23n1,τ=12n1n1n32.

Then the reflection of Vnwhich fixes the vertex iis written as σi1τσi1. Hence Dnis generated by σand 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 Gand Hbe groups. Consider the direct product set

G×HghgGhH,

and define the product of elements gh,ghG×Hto be

ghghgghh.

Then G×Hwith 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×Hthe direct product group of Gand H. Similarly, for finitely many groups G1,G2,,Gn, we can define its direct product group G1××Gn. For each 1in, if Giis a finite group of order mi, then G1××Gnis a finite group of order m1m2mn. The following theorem is famous in elementary number theory.

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

Z/mnZZ/mZ×Z/nZ.

An isomorphism f:Z/mnZZ/mZ×Z/nZis given by

xmnxmxn.

(E16) Consider the case m=2and n=3. Each element x6of Z/6Zis 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/2Zsatisfies that x+xis equal to zero. On the other hand, for the element y14Z/4Z, y+yis not equal to zero. Hence the group structures of Z/2Z×Z/2Zand Z/4Zare different.

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

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

GZ/p1e1Z××Z/prerZ.

The tuple p1e1p2e2preris 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 Ga group. For elements x,yG, if there exists some gGsuch that x=gyg1; then we say that xis conjugate to yand write xy. This is an equivalence relation on G. Namely, for any xG, we have xxby observing x=1Gx1G1. If xy, then x=gyg1for some gG. Thus y=g1xg11, and hence yx. If xyand yz, then x=gyg1and y=hzh1for some g,hG. Thus x=ghzgh1, and hence xz. For any xG, the set

CxyGyx

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

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

  1. If nis even:

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

2.      If nis odd:

1,σσ1,σ2σ2,,σn12σ1n2,τστσn1τ.

Indeed, for the case where nis 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 σiare σ±i. On the other hand, for any xDn, since

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

the conjugates of σiτare σkτfor any ksuch that kimod2. These facts induce Part (1).

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

σ=a1akb1blc1cm.

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

Elements σ,σSnare conjugate if and only if the cycle types of σand σare equal.

For example, conjugacy classes of S4are 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 Gbe a finite group. For any xG, Cxis 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 Gis a finite group. Let Vbe a finite-dimensional C-vector space. Consider the following situation. For any σGand any vV, there exists a unique element σvVsuch that

  1. σv+w=σv+σw,

  2. σαv=ασv,

  3. στv=στv,

  4. 1Gv=v

for any σ,τG, αCand v,wV. Then we say that Gacts on Vand Vis called a G-vector space.

The conditions (1) and (2) mean that for any σG, the map ρσ:VVdefined by vσvis a linear transformation on V. Furthermore, from the conditions (3) and (4), we see that for any σG, the linear transformation ρσ1is the inverse linear transformation of ρσ. Namely, each ρσis 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:GGLVby σρσ. In general, for a finite group Gand for a finite-dimensional C-vector space V, a homomorphism ρ:GGLVis called a representation of G. Then Vis a G-vector space by the action of Gon Vgiven by

σvρσv

for any σGand vV. The dimension dimCVof Vas 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 Gon Vby σvvfor any σGand vV. Namely, we can consider the homomorphism triv:GGLVby assigning σto the identity map on Vfor any σG. This is called the trivial representation of G.

(E22) For any nN, consider the cyclic group Unand the action of Unon Cgiven by the usual multiplication exp21/nzexp21/nzof the complex numbers for any kZand zC. The action of exp21/non Cis the rotation on Cin the counterclockwise direction centered at the origin with angle 2/n. If we take 1Cas a basis of the C-vector space C, we can identify GLCwith 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 S3and the numerical vector space C3. The group S3naturally acts on C3by the permutation of the components given by

σx1x2x3xσ11xσ12xσ13.

If we take the standard basis e1,e2,e3as a basis of C3, we can identify GLC3with the general linear group GL3Cby considering the matrix representation. Under this identification, the corresponding representation ρ:S3GLC3=GL3Cis given by σeσ1eσ2eσ3. Similarly, we can obtain the representation ρ:SnGLCn=GLnCthat is given by

σeσ1eσ2eσn.

This is called the permutation representation of Sn.

Next we consider subrepresentations of a representation. Let ρ:GGLVa representation. If there exists a subspace Wof Vsuch that

σwWρσwW

for any σGand wW, then Wis called a G-subspace of V. For any σG, the restriction ρσW:WWof ρσis a bijective linear transformation on W, and we obtain the representation ρW:GGLWgiven by σρσW. It is called a subrepresentation of ρ.

(E24) Consider the permutation representation ρ:S3GLC3=GL3Cas 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 ρW1is the trivial representation. Geometrically, W1and W2in C3are drawn in Figure 3. In a precise sense, if we naturally consider R3as a subset of C3, then Figure 3 shows W1R3and W2R3in R3.

Figure 3.

The subspaces W1 and W2 in C3.

For a G-vector space V, if there exist G-subspaces W1and W2of Vsuch that any element vVcan be uniquely written as

v=w1+w2w1W1w2W2,

then Vis called the direct sum of W1and W2and is written as V=W1W2. Similarly, we can define the direct sum of G-subspaces W1,W2,,Wmfor any m3. Let ρ, ρW1, and ρW2be the correspondent representations of Gto V, W1, and W2, respectively. We also say that the representation ρis the direct sum of ρW1and ρW2.

(E25) As the notation in (E24), Vis the direct sum of W1and W2. Indeed, for the standard basis e1,e2,e3of V, we see that e1+e2+e3and e1e2,e1e3are bases of W1and 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:GGLVa representation and Wa G-subspace of V. Then there exists a G-subspace Wsuch 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 Gbe a finite group and ρ:GGLVits representation. The trivial subspaces 0and Vare G-subspaces of V. If Vhas no Gsubspace other than these, Vis called the irreducible G-space, and ρis 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×=GLCis 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), ρW1is irreducible since it is one-dimensional. The representation ρW2is also irreducible. Indeed, if W2is not irreducible, there exists a one-dimensional G-subspace Win W2since W2is a 2-dimensional G-vector space. Take wW(w0). Then wis an eigenvector of ρW2σfor any σS3. However, we can see that there is no such vector in W2by direct calculations.

By observing (E25), (E26), and (E27), we see that C3is a direct sum of the irreducible G-subspaces W1and W2. In general, by using Maschke’s theorem above, we obtain.

For any representation ρ:GGLVof a finite group G, the G-vector space Vcan be written as a direct sum of some irreducible G-subspaces. Namely, ρcan 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:GGLC2be the trivial representation. Then for the standard basis e1,e2of 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:GGLV1and ρ2:GGLV2be representations of G. If there exists a bijective linear map ι:V1V2such that

ισv=σιv,σG,vV1,

then we say that V1is isomorphic to V2as a G-vector space and write V1V2. We also say that ρ1is equivalent to ρ2and write ρ1ρ2.

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

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

Let Gbe a finite group.

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

  2. For any representation ρ:GGLV, ρis 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:GGLVbe a representation. Take a basis v1,,vnof V, and fix it. By using this basis, we can consider ρσas 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,,wnis another basis of V, the matrix representation of ρσwith respect to this basis is given by P1AσPfor a some regular matrix P. Hence TrP1AσP=TrAσ. We call the map χρ:GCthe character of ρ. Remark that for elements σ,τG, if στ, then ρσρτin GLV. Thus, χρσ=χρτ. Namely, χρis constant on each of the conjugacy classes of G.

(E29) Consider the example (E25). Let ρ:S3GLC3be the permutation representation of S3. The conjugacy classes of S3are as follows:

Cycle typeConjugacy class
1111S3
21121323
3123132

Hence, in order to calculate the values of the character χρof ρ, it suffices to calculate its values on 1S3, 12, and 123. If we take the standard basis e1,e2,e3of C3, we have ρσ=eσ1eσ2eσ3, and hence

χρ1S3=3,χρ12=1,χρ123=0.

In general, as in (E29), for a representation ρ:GGLV, χρ1Gis the degree of the representation, which is equal to dimCV.

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

φψ=1GσGφσψσ¯

where z¯means the complex conjugation of zC. We call it the inner product of ϕand 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, Vcan be written as

VW1m1W2m2Wkmk

where each Wiis an irreducible G-vector space and Wiis not isomorphic to Wjas a G-vector space if ij. For each 1ik, the number miis called the multiplicity of Wiin V.

As the notation above, let ρibe the irreducible representation of Gcorrespond 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 Vis 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 ρW2are all irreducible representations of S3up to equivalence. Hence the list in (E30) is the character table of S3.

(E32) Consider the cyclic group Un. Since Unis abelian, any conjugacy class consists of a single element, and there exist nconjugacy classes. Hence there exist ndistinct 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,,ρn1are 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 σand τ, which are 1aand 1b, respectively. Next, for any 1ln1, we can consider the two-dimensional representations ρl:DnGL2Cgiven by

ρlx=cos2klπ/nsin2klπ/nsin2klπ/ncos2klπ/nifx=σk,cos2klπ/nsin2klπ/nsin2klπ/ncos2klπ/n0110ifx=σkτ.
  1. The case where nis even. For any 1ln22, since we can see χρlχρl=1by direct calculation, ρls 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,band ρlfor a,b=±1and 1ln22are all irreducible representations of Dnup to equivalence. The character table of D4is give as follows:

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

ii. The case where nis odd. Similarly, we can see that ε1,band ρlfor b=±1and 1ln12are all irreducible representations of Dnup to equivalence. The character table of D5is 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 Gand Hbe finite groups. Set

G×HghgGhH,

and define the product on G×Hby

ghghgghh.

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

In order to construct irreducible representations of G×H, we consider tensor products of vector spaces. For G-vector space Vand H-vector space W, let Fbe the vector space with basis vwvVwWand Rthe subspace of Fgenerated by

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

for any v,v1,v2V, w,w1,w2W, and αC. The quotient vector space F/Ris called the tensor product of Vand Wand is denoted by VW. The coset class of vwis denoted by vw. If v1,,vmand w1,,wnare bases of Vand W, respectively, then elements viwj(1imand 1jn) form a basis of VW. Hence dimVW=dimVdimW.

For any gGand hH, we can define the action of G×Hon VWby

ghi=1mj=1nαijviwji=1mj=1nαijgvihwj,

and hence, VWis a G×H-vector space. For the representations ρ:GGLVand ρ:GGLWcorresponding to the G-vector spaces Vand W, respectively, we denote by ρρ:GGLVWthe representation corresponding to the G×H-vector space VW. Then we have

(1) As the notation above, if ρand ρare irreducible, so is ρρ.

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

(E34) For V=Cand W=C, the tensor product VWof Vand Wis a one-dimensional C-vector space with basis 11. Thus, we have a bijective linear map VWCgiven by

a11a.

In general, we identify CCwith Cthrough 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 Γconsists 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 ieand tethe initial vertex and the terminal vertex of e. Each oriented edge ehas the inverse edge e¯such that e¯eand e¯¯=e. It is clear that ie¯=teand te¯=ie. An oriented edge esuch that ie=teis called a loop. For any v,wVΓ, we assume that there may exist more than one oriented edge whose initial vertex is vand terminal vertex w. If this is the case, we say that Γhas 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,zare depicted by small circles. The oriented edges a,b,c,d,e,f,g,hare depicted by arrows from the initial vertex to the terminal vertex, and their inverse edges are omitted for simplicity.

Figure 4.

An example of a graph.

We denote by VΓand EΓthe sets of the vertices and the oriented edges of Γ, respectively. If both VΓand EΓare finite set, we call Γa finite graph. Here, we consider only finite graphs. Remark that EΓis always even since EΓis 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 vand 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.

Figure 5.

Examples of a connected and a non-connected graph.

8.2. Automorphisms of graphs

Let Γand Γbe graphs. A morphism of directed graphs from Γto Γis 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, σmaps 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 σis bijective, then it is called an isomorphism. An isomorphism from Γto Γis called an automorphism of Γ. Let AutΓbe the set of all automorphisms of Γ. Then AutΓwith 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 Γ1consists of one vertex vand two oriented edges eand e¯. Hence all morphisms from Γ1to Γ1are automorphisms since if σ:ΓΓis a morphism, then σv=v, and σe=eor σe=e¯. If σe=e, then σe¯=e¯as a consequence, and hence σis the identity map on Γ. If σe=e¯, then σe¯=eas a consequence, and hence σis the orientation-reversing automorphism on Γ. Thus, AutΓ1=σ1σ2Z/2Zwhere σ1e=eand σ2e=e¯.

Figure 6.

Graphs which have one vertex.

On the other hand, the graph Γ2consists of one vertex vand 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 vto 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 σ5are generators of AutΓ2. In (E41), we study the structure of AutΓ2more.

Next, in order to describe the group structure of AutΓmore 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 Gbe a group, Ka subgroup of G, and Ha normal subgroup of G. Furthermore, if we have

G=hkhHkK,HK=1G,

then we call Gthe semidirect product group of Hand Kand denote it by G=HK.

(E38) Recall the dihedral group Dn=1σσ2σn1τστσ2τσn1τ. Set H1σσ2σn1and K1τ. Then we can see that the subset His a normal subgroup of Dn, HK=1, and Dn=hkhHkK. Thus Dn=HK.

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

Now, let Γbe a graph. For any v,wVΓ, we number the oriented edges of Γwith vas initial vertex and was terminal vertex. Then every oriented edge ecan be uniquely represented as e=vwk. In particular, we can arrange the numbering such that e¯=wvkfor any e=vwkEΓ.

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

Figure 7.

An example of a graph.

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

Let Tbe the subgroup of AutΓconsisting of automorphisms that fix all vertices pointwise:

TtAutΓtv=vvVΓ.

Let Mbe the subgroup of AutΓconsisting of automorphisms that fix the numberings of edges:

MmAutΓmvwk=vwkforanyvwVandanynumberk.

Then we have AutΓ=TM

(E40) Recall the graph Γ1in (E37). Since every automorphism fixes the vertex v, we see that AutΓ1=Tand M=1. Similarly, if a graph Γhas only one vertex, then AutΓ=T.

(E41) Recall the graph Γ2in (E37). We have AutΓ2=Tand M=1. Set Hσ2σ3and Kσ5. Then it is seen that HZ/2Z×Z/2Z, KZ/2Z, and AutΓ2HK.

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

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

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

Figure 8.

An example of a graph.

The subgroup Tconsists of automorphisms which permute the oriented edges e,f,g, and hence TS3. On the other hand, the subgroup Qconsists 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×Hcan be calculated with those of Gand 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.

© 2017 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Takao Satoh (December 20th 2017). Group Theory from a Mathematical Viewpoint, Symmetry (Group Theory) and Mathematical Treatment in Chemistry, Takashiro Akitsu, IntechOpen, DOI: 10.5772/intechopen.72131. Available from:

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