Linear Algebra, Theory And Applications, 2012a

F.6. NORMAL SUBGROUPS 469
If H is a subgroup, then it is also the Galois group
H = G (K, K H ) .
By Proposition F.4.7, each of these intermediate fields L is also a normal extension of F.
Now there is also something called a normal subgroup which will end up corresponding with
these normal field extensions consisting of the intermediate fields between F and K.
F.6 Normal Subgroups
When you look at groups, one of the first things to consider is the notion of a normal
subgroup.
Definition F.6.1 Let G be a group. Then a subgroup N is said to be a normal subgroup if
whenever α ∈ G,
α −1 Nα ⊆ N
The important thing about normal subgroups is that you can define the quotient group
G/N.
Definition F.6.2 Let N beasubgroupofG. 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 −1 α = ι ∈ N since
N is a subgroup. If α ∼ β, then α −1 β ∈ N and so, since N is a subgroup,
(
α −1 β ) −1
= β −1 α ∈ N
which shows that β ∼ α.
subgroup,
Now suppose αα ∼ β, then α −1 β ∈ N and so, since N is a
(
α −1 β ) −1
= β −1 α ∈ N
which shows that β ∼ α. Now suppose α ∼ β and β ∼ γ. Then α −1 β ∈ 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] the
equivalence class determined by α.
Now in the case of N a normal subgroup, you can consider the quotient group.
Definition F.6.3 Let N be a normal subgroup of a group G and define G/N as the set of
all equivalence classes with respect to the above equivalence relation. Also define
[α][β] ≡ [αβ]
Proposition F.6.4 The above definition is well defined and it also makes G/N into a
group.
Proof: First consider the claim that the definition is well defined. Suppose then that
α ∼ α ′ and β ∼ β ′ . It is required to show that
[αβ] = [ α ′ β ′]