remaining element z and make a class. If the remaining element is only z in this
moment, z can make a class by itself (as in the case of e). Notice that for an Abelian
group every element makes a class by itself. Thus, with a finite group we have a
decomposition such that
ℊ ¼ ∁ e þ ∁ a þ ∁ b þ Á Á Á þ ∁ z :
ð16:9Þ
To show that (16.9) is really a decomposition, suppose that for instance a set ∁ a \ ∁ b
is not an empty set and that x 2 ∁ a \ ∁ b . Then we must have α and β that satisfy a
following relation: x ¼ αaα
À1
¼ βbβ
À1 , i.e., b ¼ β
À1
αaα
À1
β ¼ β
À1
αa(β
À1
α)
À1 .
This implies that b has already been included in ∁ a , in contradiction to the supposition. Thus, (16.9) is in fact a decomposition of ℊ into a finite number of classes.
In the above we thought of a class conjugate to a single element. This notion can
be extended to a class conjugate to a subgroup. Let H be a subgroup of ℊ. Let g be
an element of ℊ. Let us now consider a set H
0
¼ gH g
À1
. The set H
0 is a subgroup of
ℊ and is called a conjugate subgroup.
In fact, let h i and h j be any two elements of H , that is, let gh i g
À1 and gh j g
À1 be any
tow elements of H
0 . Then, we have
gh i g
À1
À
Á gh j g
À1
À
Á ¼ gh i h j g
À1
¼ gh k g
À1 ,
ð16:10Þ
where h k ¼ h i h j 2 H . Hence, gh k g
À1
2 H
0 . Meanwhile, gh i g
À1
ð
Þ
À1 ¼ gh
À1
i g
À1
2
H
0
. Thus, conditions (1) and (2) of Sect. 16.2 are satisfied with H
0
. Therefore, H
0 is a
subgroup of ℊ. The subgroup H
0 has a same order as H . This is because with any
two different elements h i and h j gh i g
À1
6 ¼ gh j g
À1 .
If for
8
g2ℊ and a subgroup H we have a following equality
g
À1
H g ¼ H ,
ð16:11Þ
such a subgroup H is said to be an invariant subgroup. If (16.11) holds, H should be
a sum of classes (reader, please show this). A set comprising only the identity, i.e.,
{e} forms a class. Therefore, if H is a proper subgroup, H must contain two or more
classes. The relation (16.11) can be rewritten as
gH ¼ H g:
ð16:12Þ
This implies that the left coset is identical to the right coset. Thus, as far as we are
dealing with a coset pertinent to an invariant subgroup, we do not have to distinguish
left and right cosets.
Now let us anew consider the (left) coset decomposition of ℊ by an invariant
subgroup H
626
16 Introductory Group Theory
moment, z can make a class by itself (as in the case of e). Notice that for an Abelian
group every element makes a class by itself. Thus, with a finite group we have a
decomposition such that
ℊ ¼ ∁ e þ ∁ a þ ∁ b þ Á Á Á þ ∁ z :
ð16:9Þ
To show that (16.9) is really a decomposition, suppose that for instance a set ∁ a \ ∁ b
is not an empty set and that x 2 ∁ a \ ∁ b . Then we must have α and β that satisfy a
following relation: x ¼ αaα
À1
¼ βbβ
À1 , i.e., b ¼ β
À1
αaα
À1
β ¼ β
À1
αa(β
À1
α)
À1 .
This implies that b has already been included in ∁ a , in contradiction to the supposition. Thus, (16.9) is in fact a decomposition of ℊ into a finite number of classes.
In the above we thought of a class conjugate to a single element. This notion can
be extended to a class conjugate to a subgroup. Let H be a subgroup of ℊ. Let g be
an element of ℊ. Let us now consider a set H
0
¼ gH g
À1
. The set H
0 is a subgroup of
ℊ and is called a conjugate subgroup.
In fact, let h i and h j be any two elements of H , that is, let gh i g
À1 and gh j g
À1 be any
tow elements of H
0 . Then, we have
gh i g
À1
À
Á gh j g
À1
À
Á ¼ gh i h j g
À1
¼ gh k g
À1 ,
ð16:10Þ
where h k ¼ h i h j 2 H . Hence, gh k g
À1
2 H
0 . Meanwhile, gh i g
À1
ð
Þ
À1 ¼ gh
À1
i g
À1
2
H
0
. Thus, conditions (1) and (2) of Sect. 16.2 are satisfied with H
0
. Therefore, H
0 is a
subgroup of ℊ. The subgroup H
0 has a same order as H . This is because with any
two different elements h i and h j gh i g
À1
6 ¼ gh j g
À1 .
If for
8
g2ℊ and a subgroup H we have a following equality
g
À1
H g ¼ H ,
ð16:11Þ
such a subgroup H is said to be an invariant subgroup. If (16.11) holds, H should be
a sum of classes (reader, please show this). A set comprising only the identity, i.e.,
{e} forms a class. Therefore, if H is a proper subgroup, H must contain two or more
classes. The relation (16.11) can be rewritten as
gH ¼ H g:
ð16:12Þ
This implies that the left coset is identical to the right coset. Thus, as far as we are
dealing with a coset pertinent to an invariant subgroup, we do not have to distinguish
left and right cosets.
Now let us anew consider the (left) coset decomposition of ℊ by an invariant
subgroup H
626
16 Introductory Group Theory