16.2 Subgroups
As we think of subspaces in a linear vector space, we have subgroups in a group. The
definition of the subgroup is that a subset H of a group makes a group with respect to
the multiplication ⋄ that is defined for the group ℊ. The identity element makes a
group by itself. Both {e} and ℊ are subgroups as well. We often call subgroups other
than {e} and ℊ “proper” subgroups.
A necessary and sufficient condition for the subset H to be a subgroup is the
following:
(1) h i , h j 2 H ⟹h i ⋄h j 2 H .
(2) h 2 H ⟹h
À1
2 H .
If H is a subgroup of ℊ, it is obvious that the relations (1) and (2) hold.
Conversely, if (1) and (2) hold, H is a subgroup. In fact, (1) ensures the aforementioned relation (A1). Since H is a subset of ℊ, this guarantees the associative law
(A2). The relation (2) ensures (A4). Finally, in virtue of (1) and (2), h ⋄ h
À1
¼ e is
contained in H ; this implies that (A3) is satisfied. Thus, H is a subgroup, because H
satisfies the axioms (A1) to (A4). Of the above examples, (iii) has a subgroup
H ¼ e, a
2
È
É
.
It is important to decompose a set into subsets that do not mutually contain an
element (except for a special element) among them. We saw this in Part III when we
decomposed a linear vector space into subspaces. In that case the said special
element was a zero vector. Here let us consider a related question in its similar
aspects.
Let H ¼ h 1 e, h 2 , Á Á Á, h s
f
g be a subgroup of ℊ. Also let us consider aH where
∃
a 2 ℊ and a= 2H . Suppose that aH is a subset of ℊ such that aH ¼
ah 1 , ah 2 , Á Á Á, ah s
f
g . Then, we have another subset H þ aH . If H contains
s elements, so does aH . In fact, if it were not the case, namely, if ah i ¼ ah j ,
multiplying the both sides by a
À1 we would have h i ¼ h j , in contradiction. Next,
let us take b such that b= 2H and b= 2aH and make up bH and H þ aH þ bH
successively. Our question is whether these procedures decompose ℊ into subsets
mutually exclusive and collectively exhaustive.
Suppose that we can succeed in such a decomposition and get
ℊ ¼ g 1 H þ g 2 H þ Á Á Á þ g k H ,
ð16:2Þ
where g 1 , g 2 , Á Á Á, g k are mutually different elements with g 1 being the identity e. In
that case (16.2) is said to be the left coset decomposition of ℊ by H . Similarly, right
coset decomposition can be done to give
ℊ ¼ H g 1 þ H g 2 þ Á Á Á þ H g k :
ð16:3Þ
In general, however,
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16 Introductory Group Theory
As we think of subspaces in a linear vector space, we have subgroups in a group. The
definition of the subgroup is that a subset H of a group makes a group with respect to
the multiplication ⋄ that is defined for the group ℊ. The identity element makes a
group by itself. Both {e} and ℊ are subgroups as well. We often call subgroups other
than {e} and ℊ “proper” subgroups.
A necessary and sufficient condition for the subset H to be a subgroup is the
following:
(1) h i , h j 2 H ⟹h i ⋄h j 2 H .
(2) h 2 H ⟹h
À1
2 H .
If H is a subgroup of ℊ, it is obvious that the relations (1) and (2) hold.
Conversely, if (1) and (2) hold, H is a subgroup. In fact, (1) ensures the aforementioned relation (A1). Since H is a subset of ℊ, this guarantees the associative law
(A2). The relation (2) ensures (A4). Finally, in virtue of (1) and (2), h ⋄ h
À1
¼ e is
contained in H ; this implies that (A3) is satisfied. Thus, H is a subgroup, because H
satisfies the axioms (A1) to (A4). Of the above examples, (iii) has a subgroup
H ¼ e, a
2
È
É
.
It is important to decompose a set into subsets that do not mutually contain an
element (except for a special element) among them. We saw this in Part III when we
decomposed a linear vector space into subspaces. In that case the said special
element was a zero vector. Here let us consider a related question in its similar
aspects.
Let H ¼ h 1 e, h 2 , Á Á Á, h s
f
g be a subgroup of ℊ. Also let us consider aH where
∃
a 2 ℊ and a= 2H . Suppose that aH is a subset of ℊ such that aH ¼
ah 1 , ah 2 , Á Á Á, ah s
f
g . Then, we have another subset H þ aH . If H contains
s elements, so does aH . In fact, if it were not the case, namely, if ah i ¼ ah j ,
multiplying the both sides by a
À1 we would have h i ¼ h j , in contradiction. Next,
let us take b such that b= 2H and b= 2aH and make up bH and H þ aH þ bH
successively. Our question is whether these procedures decompose ℊ into subsets
mutually exclusive and collectively exhaustive.
Suppose that we can succeed in such a decomposition and get
ℊ ¼ g 1 H þ g 2 H þ Á Á Á þ g k H ,
ð16:2Þ
where g 1 , g 2 , Á Á Á, g k are mutually different elements with g 1 being the identity e. In
that case (16.2) is said to be the left coset decomposition of ℊ by H . Similarly, right
coset decomposition can be done to give
ℊ ¼ H g 1 þ H g 2 þ Á Á Á þ H g k :
ð16:3Þ
In general, however,
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16 Introductory Group Theory