g k H 6 ¼ H g k or g k H g
À1
k 6 ¼ H :
ð16:4Þ
Taking the case of left coset as an example, let us examine whether different cosets
mutually contain a common element. Suppose that g i H and g j H mutually contain a
common element. Then, that element would be expressed as g i h p ¼ g j h q (1
i,
j
n; 1
p, q
s). Thus, we have g i h p h
À1
q ¼ g j . Since H is a subgroup of ℊ,
h p h
À1
q 2 H . This implies that g j 2 g i H . It is in contradiction to the definition of left
coset. Thus, we conclude that different cosets do not mutually contain a common
element.
Suppose that the order of ℊ and H is n and s, respectively. Different k cosets
comprise s elements individually and different cosets do not mutually possess a
common element and, hence, we must have
n ¼ sk,
ð16:5Þ
where k is called an index of H . We will have many examples afterward.
16.3 Classes
Another method to decompose a group into subsets is called conjugacy classes. A
conjugate element is defined as follows: Let a be an element arbitrarily chosen from
a group. Then an element gag
À1 is called a conjugate element or conjugate to a. If
c is conjugate to b and b is conjugate to a, then c is conjugate to a. It is because
c ¼ g
0
bg
0 À1 , b ¼ gag
À1
⟹c ¼ g
0
bg
0 À1 ¼ g
0
gag
À1
g
0 À1 ¼ g
0
ga g
0
g
ð Þ
À1 : ð16:6Þ
In the above a set containing a and all the elements conjugate to a is said to be a
(conjugate) class of a. Denoting this set by ∁ a , we have
∁ a ¼ a, g 2 ag
À1
2 , g 3 ag
À1
3 , Á Á Á, g n ag
À1
n
È
É :
ð16:7Þ
In ∁ a a same element may appear repeatedly. It is obvious that in every group the
identity element e forms a class by itself. That is,
∁ e ¼ e
f g:
ð16:8Þ
As in the case of the decomposition of a group into (left or right) cosets, we can
decompose a group to classes. If group elements are not exhausted by a set
comprising ∁ e or ∁ a , let us take b such that b 6 ¼ e and b =
2 ∁ a and make ∁ b similarly
to (16.7). Repeating this procedure, we should be able to decompose a group into
classes. In fact, if group elements have not yet exhausted after these procedures, take
16.3 Classes
625
À1
k 6 ¼ H :
ð16:4Þ
Taking the case of left coset as an example, let us examine whether different cosets
mutually contain a common element. Suppose that g i H and g j H mutually contain a
common element. Then, that element would be expressed as g i h p ¼ g j h q (1
i,
j
n; 1
p, q
s). Thus, we have g i h p h
À1
q ¼ g j . Since H is a subgroup of ℊ,
h p h
À1
q 2 H . This implies that g j 2 g i H . It is in contradiction to the definition of left
coset. Thus, we conclude that different cosets do not mutually contain a common
element.
Suppose that the order of ℊ and H is n and s, respectively. Different k cosets
comprise s elements individually and different cosets do not mutually possess a
common element and, hence, we must have
n ¼ sk,
ð16:5Þ
where k is called an index of H . We will have many examples afterward.
16.3 Classes
Another method to decompose a group into subsets is called conjugacy classes. A
conjugate element is defined as follows: Let a be an element arbitrarily chosen from
a group. Then an element gag
À1 is called a conjugate element or conjugate to a. If
c is conjugate to b and b is conjugate to a, then c is conjugate to a. It is because
c ¼ g
0
bg
0 À1 , b ¼ gag
À1
⟹c ¼ g
0
bg
0 À1 ¼ g
0
gag
À1
g
0 À1 ¼ g
0
ga g
0
g
ð Þ
À1 : ð16:6Þ
In the above a set containing a and all the elements conjugate to a is said to be a
(conjugate) class of a. Denoting this set by ∁ a , we have
∁ a ¼ a, g 2 ag
À1
2 , g 3 ag
À1
3 , Á Á Á, g n ag
À1
n
È
É :
ð16:7Þ
In ∁ a a same element may appear repeatedly. It is obvious that in every group the
identity element e forms a class by itself. That is,
∁ e ¼ e
f g:
ð16:8Þ
As in the case of the decomposition of a group into (left or right) cosets, we can
decompose a group to classes. If group elements are not exhausted by a set
comprising ∁ e or ∁ a , let us take b such that b 6 ¼ e and b =
2 ∁ a and make ∁ b similarly
to (16.7). Repeating this procedure, we should be able to decompose a group into
classes. In fact, if group elements have not yet exhausted after these procedures, take
16.3 Classes
625