Group Theory from a Mathematical Viewpoint Group Theory from a Mathematical Viewpoint

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.


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.

Notation and conventions
In this section, we fix some notation and conventions and review some definitions in the set theory and the linear algebra: N ≔ the set of natural numbers ¼ 1; 2; 3; … f g Z ≔ the set of integers ¼ 0; AE1; AE2; AE3; … f g Q ≔ the set of rational numbers R ≔ the set of real numbers

Symmetry (Group Theory) and Mathematical Treatment in Chemistry
• For any a, b ∈ Z\ 0 f g, the greatest common divisor of a and b is denoted by gcd a; b ð Þ.
• 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\Y≔ x j x ∈ X; x∉Y f g .
• A map f : X ! Y is surjective if for any y ∈ Y; there exists some x ∈ X such that f x ð Þ ¼ y. • • A map f : X ! Y is bijective if f is surjective and injective. In other words, the bijective map is one-to-one correspondence between X and Y.
for any x, y ∈ V and k ∈ K.
• A linear map f : V ! V is called a linear transformation on V.

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.

Groups
Let G be a set. For any σ, τ ∈ G, if there exists the unique element σ Á τ ∈ G, 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: • (Unit) There exists some element e ∈ G such that for any σ ∈ G, We call the element e the unit of G. According to the mathematical convention, we write 1 G or simply 1, for the unit.
• (Inverse element) For any σ ∈ G, there exists some element σ 0 ∈ G such that We call σ 0 the inverse element of σ and write σ À1 . If the definition of the product is clear from the content, we often omit the symbol Á and write στ instead of σ Á τ 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 n ∈ Z, the inverse of n is Àn. 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 Â ≔R 0 f g with the usual multiplication of real numbers forms a group. We see that the unit is 1 and for any r ∈ R Â , 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 Â ≔Q 0 f g and C Â ≔C 0 f g are multiplicative groups.
(E3) For any n ∈ N n ≥ 1 ð Þ , let U n be the set of nth power roots of unity: Then U n with the usual multiplication of C forms a group. Geometrically, U n is the set of vertices of the regular n-gon on the unit circle in the complex plane C. For example, U 6 consists of the following points for ζ ¼ exp 2π ffiffiffiffi ffi À1 p 6 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 U n 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 M 2; K ð Þthe set of 2 Â 2 matrices with all entries in K: Furthermore, we denote by GL 2; K ð Þ the set of elements of M 2; K ð Þ whose determinant is not equal to zero: Then M 2; K ð Þwith the usual addition of matrices forms an additive group. The unit of M 2; K ð Þis zero matrix, and for any A ¼ a ij À Á ∈ M 2; K ð Þ, its inverse is ÀA≔ Àa ij À Á . Since GL 2; K ð Þ does not have the zero matrix, the set GL 2; K ð Þ is not an additive group. On the other hand, the set GL 2; K ð Þ with the usual multiplication of matrices forms a multiplicative group. The unit of GL 2; K ð Þ is the unit matrix E 2 , and for any A ¼ a ij À Á ∈ GL 2; K ð Þ, its inverse is the inverse matrix A À1 as follows: Àa 12 Àa 21 a 11 : The group GL 2; K ð Þ is called the general linear group of degree 2. Similarly, we can consider the general linear group GL n; K ð Þof degree n for any n ∈ N.
Both M 2; K ð Þ and GL 2; K ð Þ are infinite groups. But the most significant difference between them is the commutativity of the products. Although we see For a group G, if στ ¼ τσ holds for any σ, τ ∈ G, then G is called an abelian group. The group GL 2; K ð Þis a non-abelian group, and all the groups as mentioned before except for GL 2; K ð Þare abelian groups.

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, τ ∈ H, στ ∈ H.
We can consider H itself is a group by restricting the product of G to H. For any group G, the one point subset 1 G f g 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 n ∈ Z, the subset nZ≔ 0; AEn; AE2n; … f g ⊂ Z of Z consisting of multiples of n is a subgroup of Z. Since 0Z ¼ 0 f g is the trivial subgroup, and since nZ ¼ Àn ð ÞZ, we usually consider the case n ∈ N.
(E6) Consider the group U 6 consisting of 6th power roots of unity. Then we can consider U 2 and U 3 are subgroups of U 6 .
(E7) Let K be Q, R, or C. The subset Þ consisting of matrices whose determinants are equal to one is a subgroup of GL 2; K ð Þ. We call SL 2; K ð Þthe 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 S h i≔ s e1 1 s e2 2 ⋯s em m j m ∈ Z ≥ 0 ; s i ∈ S; e i ¼ AE1

È É
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 ¼ 0, the product s e1 1 ⋯s em m means 1 G and that for any σ ¼ s e1 1 s e2 2 ⋯s em m ∈ S h i, its inverse is given by σ À1 ¼ s Àem m s ÀemÀ1 mÀ1 ⋯s Àe1 1 . We call S h i the subgroup of G generated by S. The elements of S are called generators of the subgroup S h i. Here we give some examples: (E8) The additive group Z is generated by 1. For any n ≥ 1, the group U n of nth power roots of unity is generated by In general, a group generated by a single element is called a cyclic group. Thus, Z is an infinite cyclic group, and U n is a finite cyclic group. Remark that À1 and ζ À1 ¼ exp À2π ffiffiffiffiffiffi ffi À1 p =n À Á are also generators of Z and U n , respectively.
(E9) It is known that the additive groups Q, R, and C and the multiplicative groups GL 2; K ð Þ and SL 2; K ð Þfor K ¼ 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 σ ∈ G, the subset σH≔ στ ∈ G jτ ∈ H f g is called a left coset of H in G. We can see that σH ¼ τH if and only if there exists some h ∈ H such that σ ¼ τh.
(E10) In the additive group Z, for any n ∈ N, consider the subgroup nZ. Then, since the product of Z is written additively, a left coset of nZ is given by On the other hand, if we take the remainder r of the division of σ by n, then we see σ þ nZ ¼ r þ nZ. Hence all left cosets of nZ in Z are given by ÞþnZ: For simplicity, we write r ½ n for r þ nZ.
(E11) Consider the finite cyclic group U 6 and its subgroup Then we can see that Hence there exist three left cosets of U 2 .
In example (E11), we can see that the order of U 2 times the number of left cosets of U 2 is equal to six, which is the order of U 6 . 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 ½ . Then we have the following: Theorem 3.1 (Lagrange). As the above notation C, we have |G| ¼ |H| G : H ½ . 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.

Quotient groups
For a group G and its subgroup H, the set of left cosets of H is denoted by 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 σnσ À1 ∈ N for any n ∈ N and any σ ∈ G, 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, τN ∈ G=N, define σN Á τN≔ στ ð ÞN: Then this definition is well defined, and G=N with this product forms a group. The unit is 1 G N ¼ N, and for any σN ∈ G=N, its inverse is given by σN ð Þ À1 ¼ σ À1 N. We call G=N the quotient group of G by N.
(E12) The most important example for quotient groups is For example, in the group Z=6Z, we have For any 0 ≤ r ≤ n À 1, since we see the group Z=nZ is a cyclic group of order n generated by 1 ½ n .

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 : 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 G ffi H.
and consider it as a multiplicative subgroup of R Â . The exponent map exp : R ! R >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 GL 2; K ð Þ!K Â is a homomorphism. It is, however, not an isomorphism since f is not injective. For example, det On the other hand, SL 2; K ð Þ is a normal subgroup of GL 2; K ð Þ. For any σ, τ ∈ GL 2; K ð Þ, we can see that Define the map f : Then f is an isomorphism. Indeed f is injective. For any x ∈ K Â , if we consider the element Þ¼x. Hence f is surjective. Moreover, we have 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.

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

Symmetric groups
For any n ∈ N, set X≔ 1; 2; …; n f g . A bijective map σ : X ! X is called a permutation on X. A permutation σ is denoted by Remark that this is not a matrix. We can omit a letter i 1 ≤ i ≤ n ð Þif the letter i is fixed. For example, for n ¼ 4: We call the permutation Let S n be the set of permutations on X. For any σ, τ ∈ S n , define the product of σ and τ to be the composition σ ∘ τ as a map. Then the set S n with this product forms a group. We call it the symmetric group of degree n. The unit is the identity permutation, and for any σ ∈ S n , its inverse is given by The symmetric group S n is a finite group of order n!.
Since S 1 is the trivial group, and Hence, S 3 is non-abelian. Similarly, for any n ≥ 3, S n is non-abelian.
Here we consider another description of permutations. For distinct letters a 1 , …, a m ∈ X, the permutation is denoted by a 1 ; a 2 ; ⋯; a m ð Þand is called a cyclic permutation of length m. We call a cyclic permutation of length 2 a transposition. Namely, any transposition is of type A cyclic permutation of length 1 is nothing but the identity permutation: 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 ¼ 1 2 3 4 5 3 5 4 1 2 : Then we see  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 a 1 ; a 2 ; ⋯; a m ð Þis even (resp. odd) permutation if m is odd (resp. even).

Alternating groups
In this subsection, we consider important normal subgroups of the symmetric groups. Let A n be the set of even permutations of S n . For any σ ∈ A n , if we write σ as a product of transpositions, σ ¼ τ 1 ⋯τ k , then we see Clearly, if σ, τ ∈ A n , then στ ∈ A n . Thus, the subset A n is a subgroup of S n . We call A n the alternating group of degree n. It is easily seen that A n is a normal subgroup of S n . For example, for n ¼ 3 and 4, we have A 3 ¼ ε; For any σ ∈ S n , we have if σ is even permutation: & Hence S n : A n ½ ¼2. Therefore, from Lagrange's theorem, we see that A n is a finite group of order n!=2.

Dihedral groups
For any n ∈ N n ≥ 3 ð Þ , consider a regular polygon V n with n sides, and fix it. A map σ : V n ! V n is called a congruent transformation on V n if σ preserves the distance between any two points in V n . Namely, σ is considered as a symmetry on V n . Set D n ≔ σ : V n ! V n j σ is a congruent transformation f g : For any σ, τ ∈ D n , define the product of σ and τ to be the composition σ ∘ τ as a map. Then the set D n 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 V n is determined by the correspondence between vertices of V n . Indeed, attach the number 1, 2, …, n to vertices of V n in counterclockwise direction. For any σ ∈ D n , if σ 1 ð Þ ¼ i, then the vertices 2, 3, …, n are mapped to i þ 1, i þ 2, …, n, 1, 2, …i À 1, respectively, Cor mapped to i À 1, i À 2, …, 1, n, n À 1, …, i þ 1, respectively. If we express this by using the notation for permutations, we have The former case is a rotation, and the latter case is the composition of a rotation and a reflection. For n ¼ 3, see Figure 2. Thus the dihedral group D n is a finite group of order 2n and is naturally considered as a subgroup of S n . For n ¼ 3, since D 3 is a subgroup of S 3 , and since both groups are of order 6, we see that Let σ ∈ D n be the rotation of V n with angle 2π n in the counterclockwise direction and τ ∈ D n be the reflection of V n which fixes the vertex 1. Namely, Then the reflection of V n which fixes the vertex i is written as σ iÀ1 τσ À iÀ1 ð Þ . Hence D n is generated by σ and τ. Moreover, we have D n ¼ 1; σ; σ 2 ; …; σ nÀ1 ; τ; στ; σ 2 τ; …; σ nÀ1 τ È É :

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 and define the product of elements g; h ð Þ, g 0 ; h 0 ð Þ∈ G Â H to be g; h ð ÞÁ g 0 ; h 0 ð Þ≔ gg 0 ; hh 0 ð Þ : Then G Â H with this product forms a group. The unit is 1 G ; 1 H ð Þ, and for any g; h ð Þ∈ G Â H, its inverse is given by g À1 ; h À1 À Á ∈ G Â H. We call the group G Â H the direct product group of G and H. Similarly, for finitely many groups G 1 , G 2 , …, G n , we can define its direct product group G 1 Â ⋯ Â G n . For each 1 ≤ i ≤ n, if G i is a finite group of order m i , then G 1 Â ⋯ Â G n is a finite group of order m 1 m 2 ⋯m n . The following theorem is famous in elementary number theory. An isomorphism f : Z=mnZ ! Z=mZ Â Z=nZ is given by Group Theory from a Mathematical Viewpoint http://dx.doi.org/10.5772/intechopen.72131 (E16) Consider the case m ¼ 2 and n ¼ 3. Each element x ½ 6 of Z=6Z is mapped to the following element by the above isomorphism f : (E17) If gcd m; n ð Þ6 ¼ 1, the theorem does not hold. For example, consider the case of m ¼ n ¼ 2. Any element x ∈ Z=2Z Â Z=2Z satisfies that x þ x is equal to zero. On the other hand, for the element y≔ 1 ½ 4 ∈ Z=4Z, y þ y is not equal to zero. Hence the group structures of Z=2Z Â Z=2Z and Z=4Z are different. Now, we show one of the most important theorems in finite group theory.
Theorem 4.4 (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: The tuple p e1 1 ; p e2 2 ; …; p er r À Á 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

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, y ∈ G, if there exists some g ∈ G such that x ¼ gyg À1 ; then we say that x is conjugate to y and write x $ y. This is an equivalence relation on G. Namely, for any x ∈ G, we have x $ x by observing x ¼ 1 G x1 À1 G . If x $ y, then x ¼ gyg À1 for some g ∈ G. Thus y ¼ g À1 x g À1 À Á À1 , and hence y $ x. If x $ y and y $ z, then x ¼ gyg À1 and y ¼ hzh À1 for some g, h ∈ G. Thus x ¼ gh ð Þz gh ð Þ À1 , and hence x $ z. For any x ∈ G, the set C x ð Þ≔ y ∈ Gj y $ x f g is called the conjugacy class of x in G. If G is abelian group, for any x ∈ G, there exists no element conjugate to x except for x, and hence C x ð Þ ¼ x f g. Here we give a few examples.
(E19) (Dihedral groups) For n ≥ 3, the conjugacy classes of D n are as follows: 1. If n is even: If n is odd: Indeed, for the case where n is even, we can see the above from the following observation. For any x ∈ D n , since ( the conjugates of σ i are σ AEi . On the other hand, for any x ∈ D n , since ( the conjugates of σ i τ are σ k τ for any k such that k i mod2 ð Þ. These facts induce Part (1).
(E20) (Symmetric groups) For any σ ∈ S n , we can write σ as a product of cyclic permutations which do not have a common letter, like Furthermore, we may assume k ≥ l ≥ ⋯ ≥ m since the cyclic permutations appeared in the right hand side are commutative. Then we call k; l; …; m ð Þis the cycle type of σ.

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.

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 σ ∈ G and any v ∈ V, there exists a unique element σ Á v ∈ V such that for any σ, τ ∈ G, α ∈ C and v, w ∈ V. 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 σ ∈ G, the map r σ ð Þ : V ! V defined by v ↦ σ Á v is a linear transformation on V. Furthermore, from the conditions (3) and (4), we see that for any σ ∈ G, the linear transformation r σ À1 À Á is the inverse linear transformation of r σ ð Þ. Namely, each r σ ð Þ is a bijective. Set and consider it as a group with the product given by the composition of maps. Then we obtain the group homomorphism r : G ! GL V ð Þ by σ ↦ r σ ð Þ. In general, for a finite group G and for a finite-dimensional C-vector space V, a homomorphism r : G ! GL V ð Þ is called a representation of G. Then V is a G-vector space by the action of G on V given by for any σ ∈ G and v ∈ V. The dimension dim C V of V as a C-vector space is called the degree of the representation r. 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 σ Á v≔v for any σ ∈ G and v ∈ V. Namely, we can consider the homomorphism triv : G ! GL V ð Þ by assigning σ to the identity map on V for any σ ∈ G. This is called the trivial representation of G.
(E22) For any n ∈ N, consider the cyclic group U n and the action of U n on C given by the usual multiplication exp 2kπ ffiffiffiffiffiffi ffi À1 p =n À Á Á z≔exp 2kπ ffiffiffiffiffiffi ffi À1 p =n À Á z of the complex numbers for any k ∈ Z and z ∈ C. The action of exp 2kπ ffiffiffiffiffiffi ffi À1 p =n À Á on C is the rotation on C in the counterclockwise direction centered at the origin with angle 2kπ ð Þ=n. If we take 1 ∈ C as a basis of the C-vector space C, we can identify GL C ð Þ with the general linear group GL 1; C ð Þ¼C Â by considering the matrix representation. Under this identification, the corresponding representation r : U n ! GL C ð Þ ¼ C Â is given by the natural inclusion map U n´C Â .
(E23) Consider the symmetric group S 3 and the numerical vector space C 3 . The group S 3 naturally acts on C 3 by the permutation of the components given by If we take the standard basis e 1 , e 2 , e 3 as a basis of C 3 , we can identify GL C 3 À Á with the general linear group GL 3; C ð Þ by considering the matrix representation. Under this identification, the corresponding representation r : Similarly, we can obtain the representation r : S n ! GL C n ð Þ ¼ GL n; C ð Þthat is given by This is called the permutation representation of S n .
Next we consider subrepresentations of a representation. Let r : G ! GL V ð Þ a representation. If there exists a subspace W of V such that for any σ ∈ G and w ∈ W, then W is called a G-subspace of V. For any σ ∈ G, the restriction r σ ð Þj W : W ! W of r σ ð Þ is a bijective linear transformation on W, and we obtain the representation rj W : G ! GL W ð Þ given by σ ↦ r σ ð Þj W . It is called a subrepresentation of r.
(E24) Consider the permutation representation r : Þas in (E23). Let us consider subspaces of C 3 . It is easily seen that these are S 3 -subspaces and the subrepresentation rj W1 is the trivial representation. Geometrically, W 1 and W 2 in C 3 are drawn in Figure 3. In a precise sense, if we naturally consider R 3 as a subset of C 3 , then Figure 3 shows W 1 ∩ R 3 and W 2 ∩ R 3 in R 3 .
For a G-vector space V, if there exist G-subspaces W 1 and W 2 of V such that any element v ∈ V can be uniquely written as then V is called the direct sum of W 1 and W 2 and is written as Similarly, we can define the direct sum of G-subspaces W 1 , W 2 , …, W m for any m ≥ 3. Let r, rj W1 , and rj W2 be the correspondent representations of G to V, W 1 , and W 2 , respectively. We also say that the representation r is the direct sum of rj W1 and rj W2 .
(E25) As the notation in (E24), V is the direct sum of W 1 and W 2 . Indeed, for the standard basis e 1 , e 2 , e 3 of V, we see that e 1 þ e 2 þ e 3 and e 1 À e 2 , e 1 À e 3 are bases of W 1 and W 2 , respectively.
Thus, for any x ¼ x 1 e 1 þ x 2 e 2 þ x 3 e 3 ∈ C 3 , we can rewrite Þ : Furthermore, we verify that this expression is unique by direct calculations.
In general, we have Theorem 6.1 (Maschke). Let r : G ! GL V ð Þ a representation and W a G-subspace of V. Then there exists a G-subspace W 0 such that V ¼ W ⊕ W 0 .

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 r : G ! GL V ð Þ its representation. The trivial subspaces 0 f g and V are G-subspaces of V. If V has no G subspace other than these, V is called the irreducible Gspace, and r is called the irreducible representation of G.
(E26) Any one-dimensional representation is trivial. For example, the representation r : Let us consider the other example. For any σ ∈ S n , set if σ is odd permutation: & Then we can easily see that the map sgn : S n ! C Â ¼ GL C ð Þ is a homomorphism and, hence, is a representation of S n . This irreducible representation is called the signature representation of S n .
(E27) As the notation in (E24), rj W1 is irreducible since it is one-dimensional. The representation rj W2 is also irreducible. Indeed, if W 2 is not irreducible, there exists a one-dimensional Gsubspace W in W 2 since W 2 is a 2-dimensional G-vector space. Take w ∈ W (w 6 ¼ 0). Then w is an eigenvector of rj W2 σ ð Þ for any σ ∈ S 3 . However, we can see that there is no such vector in W 2 by direct calculations.
By observing (E25), (E26), and (E27), we see that C 3 is a direct sum of the irreducible Gsubspaces W 1 and W 2 . In general, by using Maschke's theorem above, we obtain.
Theorem 6.2. For any representation r : G ! GL V ð Þ of a finite group G, the G-vector space V can be written as a direct sum of some irreducible G-subspaces. Namely, r 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 r : G ! GL C 2 À Á be the trivial representation. Then for the standard basis e 1 , e 2 of C 2 , we have In order to do the classification of representations, we consider the equivalency of representations. Let r 1 : G ! GL V 1 ð Þ and r 2 : G ! GL V 2 ð Þ be representations of G. If there exists a bijective linear map ι : then we say that V 1 is isomorphic to V 2 as a G-vector space and write V 1 ffi V 2 . We also say that r 1 is equivalent to r 2 and write r 1 $ r 2 .
(E28) For any group G. let unit : G ! GL C ð Þ ¼ C Â be the trivial representation of G. Then any trivial representation r : G ! GL V ð Þ 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. Theorem 6.3. 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 r : G ! GL V ð Þ, r is equivalent to a direct sum of some irreducible representations: Furthermore, the tuple of the components is uniquely determined by G, up to the order.

Characters
In this subsection, for a given representation, we give a method to determine whether it is irreducible or not by using characters. Let r : G ! GL V ð Þ be a representation. Take a basis v 1 , …, v n of V, and fix it. By using this basis, we can consider r σ ð Þ as an n Â n ð Þ-matrix A σ ¼ a ij À Á , which is the matrix representation of r σ ð Þ. Then set for any σ ∈ G. Remark that this definition is well defined since it does not depend on the choice of a basis of V. Indeed, if w 1 , …, w n is another basis of V, the matrix representation of r σ ð Þ with respect to this basis is given by P À1 A σ P for a some regular matrix P. Hence Tr P À1 A σ P À Á ¼ Tr A σ ð Þ. We call the map χ r : G ! C the character of r. Remark that for elements σ, τ ∈ G, if σ $ τ, then r σ ð Þ $ r τ ð Þ in GL V ð Þ. Thus, χ r σ ð Þ ¼ χ r τ ð Þ. Namely, χ r is constant on each of the conjugacy classes of G.
(E29) Consider the example (E25). Let r : S 3 ! GL C 3 À Á be the permutation representation of S 3 . The conjugacy classes of S 3 are as follows: Hence, in order to calculate the values of the character χ r of r, it suffices to calculate its values on 1 S3 , 1; 2 ð Þ, and 1; 2; 3 ð Þ. If we take the standard basis e 1 , e 2 , e 3 of C 3 , we have r σ ð Þ ¼ e σ 1 ð Þ e σ 2 ð Þ e σ 3 ð Þ À Á , and hence In general, as in (E29), for a representation r : G ! GL V ð Þ, χ r 1 G ð Þ is the degree of the representation, which is equal to dim C V ð Þ.
where z means the complex conjugation of z ∈ C. 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.
r is irreducible ⇔ χ r ; χ r ¼ 1: (E30) We have the three irreducible representations of S 3 . By direct calculations, we obtain the following list: Hence we see that in each of cases, we have χ r ; χ r ¼ 1.
By Theorem 6.3, we see that for any representation r : G ! GL V ð Þ, V can be written as where each W i is an irreducible G-vector space and W i is not isomorphic to W j as a G-vector space if i 6 ¼ j. For each 1 ≤ i ≤ k, the number m i is called the multiplicity of W i in V. Theorem 6.5. As the notation above, let r i be the irreducible representation of G correspond to the Gvector space W i . Then we have Namely, each of the multiplicity of the irreducible G-vector spaces in V is calculated by the inner product of the characters it turns out that unit, sgn , and rj W2 are all irreducible representations of S 3 up to equivalence. Hence the list in (E30) is the character table of S 3 .
(E32) Consider the cyclic group U n . Since U n 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 0 ≤ l ≤ n À 1, define the map r l : U n ! GL C ð Þ ¼ C Â by Then we obtain Hence we see that r 0 , r 1 , …, r nÀ1 are nonequivalent one-dimensional representations, and hence the above list is the character table of U n . In general, all irreducible representations of an abelian group are of degree 1.
(E33) (Dihedral groups) For n ≥ 3, consider the dihedral groups D n . First, for any a, b ¼ AE1, there exist the four one-dimensional representations ε a, b : D n ! C Â defined by ( These maps are characterized by the images of σ and τ, which are À1 ð Þ a and À1 ð Þ b , respectively. Next, for any 1 ≤ l ≤ n À 1, we can consider the two-dimensional representations r l : D n ! GL 2; C ð Þgiven by i. The case where n is even. For any 1 ≤ l ≤ nÀ2 2 , since we can see χ r l ; χ r l D E ¼ 1 by direct calculation, r l s are irreducible representations of D n . Since we have χ ε1,1 1 ð Þ 2 þ χ ε1,À1 1 ð Þ 2 þ χ εÀ1,1 1 ð Þ 2 þ χ εÀ1,À1 1 ð Þ 2 it turns out that ε a, b and r l for a, b ¼ AE1 and 1 ≤ l ≤ nÀ2 2 are all irreducible representations of D n up to equivalence. The character table of D 4 is give as follows: ii. The case where n is odd. Similarly, we can see that ε 1, b and r l for b ¼ AE1 and 1 ≤ l ≤ nÀ1 2 are all irreducible representations of D n up to equivalence. The character table of D 5 is give as follows:

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.
On the other hand, the graph Γ 2 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 Γ 2 . Namely, all of them map v to v, and the correspondences of edges are given by Hence Aut Γ 2 ð Þ ¼ σ 1 ; …; σ 8 f g . It turns out that σ 2 , σ 3 , and σ 5 are generators of Aut Γ 2 ð Þ. In (E41), we study the structure of Aut Γ 2 ð Þ more.
Next, in order to describe the group structure of Aut Γ ð Þ 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