Appendix A - Background facts about groups

Summary

We collect here, for reference, some of the standard results in group theory which have been used throughout the book. We begin with some of the standard elementary properties of p-groups and Sylow subgroups.

Lemma A.1For any pair of p-groups P ≤ Q, if N_Q(P) = P, then Q = P.

Proof. This is a general property of nilpotent groups (finite or not). If G is nilpotent and H < G, then by definition, there is K > H such that [G, K] ≤ H. Hence H < K ≤ N_G(H). That all p-groups are nilpotent is shown, e.g., in [A4, 9.8] or [G1, Theorem 2.3.3].

Lemma A.2Fix a p-group P, and an automorphism α ∈ Aut(P) of order prime to p. Assume 1 = P₀ ⊴ P₁ ⊴ … ⊴ P_m = P is a sequence of subgroups all normal in P, such that for each 1 ≤ i ≤ m,α|P_i ≡ Id_Pi (mod P_i-1). Thenα = Id_P.

Proof. See, for example, [G1, Theorem 5.3.2]. It suffices by induction to prove this when m = 2, and when the order of α is a prime q ≠ p. In this case, for each g ∈ P, α acts on the coset _gP₁ with fixed subset of order ≡ |_gP₁| (mod q). Since |_gP₁| = |P₁| is a power of p, this shows that α fixes at least one element in _gP₁. Thus α is the identity on P₁ and on at least one element in each coset of P₁, and so α = Id_P.