Elementary results

Summary

Definition. A group of automorphisms of a group P stabilizes a chain if. Here [a, x] = x^−ax for x ∈ P, a ∈ A.

Theorem 0.1. If a group of automorphisms A of a π'-group P stabilizes a chain, then A is a π-group.

Proof. Suppose a ∈ A is a π-automorphism of P. Clearly, by induction we may assume that [a, P₁] = 1. Then if x ∈ P, x^a = xy where y ∈ P₁, since.

It follows that.

Since y is a π- element while a is a π'-element, we have that y = 1 and [a, P] = 1. Thus a = 1 and A is a π-group.

Corollary 0. 2. If A is a π'-group of automorphisms of a π-group P such that [P, A, A] = 1, then [P, A] = 1 and so A = 1.

Proof. A stabilizes the chain.

Lemma 0. 3. Let A be a π-group of automorphisms of a π-group P. Let Q be an A-invariant normal subgroup of P. Then C_P/Q(A) = C_P(A)Q/Q.

Proof. Clearly.

Suppose now that xQ is a coset of Q in P which is fixed by A. Let QA act as a group of permutations on xQ where A acts in the obvious way and Q acts by multiplication on the right.