is denoted by a symbol
"
⋄
" below. Note that the symbol ⋄ implies an ordinary
multiplication, an ordinary addition, etc.
(A1) If a and b are any elements of the set ℊ, then so is a ⋄ b. (We sometimes say
that the set is “closed” regarding the multiplication.)
(A2) Multiplication is associative; i.e., a ⋄ (b ⋄ c) ¼ (a ⋄ b) ⋄ c.
(A3) The set ℊ contains an element of e called the identity element such that we
have a ⋄ e ¼ e ⋄ a ¼ a with any element a of ℊ.
(A4) For any a of ℊ, we have an element b such that a ⋄ b ¼ b ⋄ a ¼ e. The element
b is said to be the inverse element of a. We denote b a
À1 .
In the above definitions, we assume that the commutative law does not necessarily hold, that is, a ⋄ b 6 ¼ b ⋄ a. In that case the group ℊ is said to be a
noncommutative group. However, we have a case where the commutative law
holds, i.e., a ⋄ b ¼ b ⋄ a. If so, the group ℊ is called a commutative group or an
Abelian group.
Let us think of some examples of groups. Henceforth, we follow the convention
and write ab to express a ⋄ b.
Example 16.1 We present several examples of groups below. Examples (i)–(iv) are
simple, but Example (v) is general.
(i) ℊ ¼ {1, À1}. The group ℊ makes a group with respect to the multiplication.
This is an example of a finite group.
(ii) ℊ ¼ {Á Á Á, À3, À2, À1, 0, 1, 2, 3, Á Á Á}. The group ℊ makes a group with respect
to the addition. This is an infinite group. For instance, take a(>0) and make
a + 1 and make (a + 1) + 1, [(a + 1) + 1] + 1, Á Á Á again and again. Thus, addition
is not closed and, hence, we must have an infinite group.
(iii) Let us start with a matrix a ¼
0 À1
1 0
. Then, the inverse a
À1
¼ a
3
¼
0 1
À1 0
. We have a
2
¼
À1 0
0 À1
. Its inverse is a
2 itself. These four
elements make a group. That is, ℊ ¼ {e, a, a
2 , a
3 }. This is an example of cyclic
groups.
(iv) ℊ ¼ {1}. It is a most trivial case, but sometimes the trivial case is very
important as well. We will come back later to this point.
(v) Let us think of a more general case. In Chap. 11, we discussed endomorphism
on a vector space and showed the necessary and sufficient condition for the
existence of an inverse transformation. In this relation, we consider a set that
comprises matrices such that
GL n, ℂ
ð
Þ A ¼ a ij
À Á ji, j ¼ 1, 2, Á Á Á, n; a ij 2 ℂ, detA 6 ¼ 0
È
É :
This may be either a finite group or an infinite group. The former can be a symmetry
group and the latter can be a rotation group. This group is characterized by a set of
622
16 Introductory Group Theory
"
⋄
" below. Note that the symbol ⋄ implies an ordinary
multiplication, an ordinary addition, etc.
(A1) If a and b are any elements of the set ℊ, then so is a ⋄ b. (We sometimes say
that the set is “closed” regarding the multiplication.)
(A2) Multiplication is associative; i.e., a ⋄ (b ⋄ c) ¼ (a ⋄ b) ⋄ c.
(A3) The set ℊ contains an element of e called the identity element such that we
have a ⋄ e ¼ e ⋄ a ¼ a with any element a of ℊ.
(A4) For any a of ℊ, we have an element b such that a ⋄ b ¼ b ⋄ a ¼ e. The element
b is said to be the inverse element of a. We denote b a
À1 .
In the above definitions, we assume that the commutative law does not necessarily hold, that is, a ⋄ b 6 ¼ b ⋄ a. In that case the group ℊ is said to be a
noncommutative group. However, we have a case where the commutative law
holds, i.e., a ⋄ b ¼ b ⋄ a. If so, the group ℊ is called a commutative group or an
Abelian group.
Let us think of some examples of groups. Henceforth, we follow the convention
and write ab to express a ⋄ b.
Example 16.1 We present several examples of groups below. Examples (i)–(iv) are
simple, but Example (v) is general.
(i) ℊ ¼ {1, À1}. The group ℊ makes a group with respect to the multiplication.
This is an example of a finite group.
(ii) ℊ ¼ {Á Á Á, À3, À2, À1, 0, 1, 2, 3, Á Á Á}. The group ℊ makes a group with respect
to the addition. This is an infinite group. For instance, take a(>0) and make
a + 1 and make (a + 1) + 1, [(a + 1) + 1] + 1, Á Á Á again and again. Thus, addition
is not closed and, hence, we must have an infinite group.
(iii) Let us start with a matrix a ¼
0 À1
1 0
. Then, the inverse a
À1
¼ a
3
¼
0 1
À1 0
. We have a
2
¼
À1 0
0 À1
. Its inverse is a
2 itself. These four
elements make a group. That is, ℊ ¼ {e, a, a
2 , a
3 }. This is an example of cyclic
groups.
(iv) ℊ ¼ {1}. It is a most trivial case, but sometimes the trivial case is very
important as well. We will come back later to this point.
(v) Let us think of a more general case. In Chap. 11, we discussed endomorphism
on a vector space and showed the necessary and sufficient condition for the
existence of an inverse transformation. In this relation, we consider a set that
comprises matrices such that
GL n, ℂ
ð
Þ A ¼ a ij
À Á ji, j ¼ 1, 2, Á Á Á, n; a ij 2 ℂ, detA 6 ¼ 0
È
É :
This may be either a finite group or an infinite group. The former can be a symmetry
group and the latter can be a rotation group. This group is characterized by a set of
622
16 Introductory Group Theory