222 CHAPTER 9. SOME ITEMS WHICH RESEMBLE LINEAR ALGEBRA

Definition 9.4.20 Let G be a group. A subset N of a group G is called a subgroup if itcontains ι the identity and is closed with respect to the operation on G. That is, if α,β ∈N,then αβ ∈ N. Then a subgroup N is said to be a normal subgroup if whenever α ∈ G,

α−1Nα ⊆ N

The important thing about normal subgroups is that you can define the quotient groupG/N.

Definition 9.4.21 Let N be a subgroup of G. Define an equivalence relation ∼ as follows.

α ∼ β means α−1

β ∈ N

Why is this an equivalence relation? It is clear that α ∼ α because α−1α = ι ∈ N sinceN is a subgroup. If α ∼ β , then α−1β ∈ N and so, since N is a subgroup,(

α−1

β)−1

= β−1

α ∈ N

which shows that β ∼ α . Now suppose α ∼ β and β ∼ γ. Then α−1β ∈ N and β−1

γ ∈ N.Then since N is a subgroup

α−1

ββ−1

γ = α−1

γ ∈ N

and so α ∼ γ which shows that it is an equivalence relation as claimed. Denote by [α] theequivalence class determined by α .

Now in the case of N a normal subgroup, you can consider the quotient group.

Definition 9.4.22 Let N be a normal subgroup of a group G and define G/N as the set ofall equivalence classes with respect to the above equivalence relation. Also define

[α] [β ]≡ [αβ ]

Proposition 9.4.23 The above definition is well defined and it also makes G/N into agroup.

Proof: First consider the claim that the definition is well defined. Suppose then thatα ∼ ᾱ and β ∼ β̄ . It is required to show that

[αβ ] =[ᾱβ̄]

Is (αβ )−1ᾱβ̄ ∈ N? Is β

−1α−1ᾱβ̄ ∈ N?

(αβ )−1ᾱβ̄ = β

−1α−1

ᾱβ̄ = β−1

∈N︷ ︸︸ ︷α−1

ᾱβ̄

=

∈N︷ ︸︸ ︷β−1 (

α−1

ᾱ)

β

∈N︷ ︸︸ ︷β−1

β̄ = n1n2 ∈ N

Thus the operation is well defined. Clearly the identity is [ι ] where ι is the identity in Gand the inverse is

[α−1

]where α−1 is the inverse for α in G. The associative law is also

obvious. ■Note that it was important to have the subgroup be normal in order to have the operation

defined on the quotient group consisting of the set of equivalence classes.