Milne - Group Theory.. - Free

32 3 ISOMORPHISM THEOREMS. EXTENSIONS.
Proof. Let Q = C G (N). We shall check that N and Q satisfy the conditions of
Proposition 3.6.
Observe first that, for any g ∈ G, n ↦→ gng −1 : N → N is an automorphism of
N, and (because N is complete), it must be the inner automorphism defined by an
element γ = γ(g) ofN; thus
gng −1 = γnγ −1 all n ∈ N.
This equation shows that γ −1 g ∈ Q, and hence g = γ(γ −1 g) ∈ NQ. Since g was
arbitrary, we have shown that G = NQ.
Next note that every element of N ∩ Q is in the centre of N, wh ich (by th e
completeness assumption) is trivial; hence N ∩ Q =1.
Finally, for any element g = nq ∈ G,
gQg −1 = n(qQq −1 )n −1 = nQn −1 = Q
(recall that every element of N commutes withevery element of Q). Therefore Q is
normal in G.
An extension
1 → N → G → Q → 1
gives rise to a homomorphism θ ′ : G → Aut(N), namely,
θ ′ (g)(n) =gng −1 .
Let ˜q ∈ G map to q in Q; then the image of θ ′ (˜q) inAut(N)/Inn(N) depends only
on q; therefore we get a homomorphism
θ : Q → Out(N) df
=Aut(N)/Inn(N).
This map θ depends only on the isomorphism class of the extension, and we write
Ext 1 (G, N) θ for the set of isomorphism classes of extensions with a given θ. These
sets have been extensively studied.
The Hölder program.
Recall that a group G is simple if it contains no normal subgroup except 1 and G.
In other words, a group is simple if it can’t be realized as an extension of smaller
groups. Every finite group can be obtained by taking repeated extensions of simple
groups. Thus the simple finite groups can be regarded as the basic building blocks
for all finite groups.
The problem of classifying all simple groups falls into two parts:
A. Classify all finite simple groups;
B. Classify all extensions of finite groups.