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 NQ(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 ≤ NG(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 ⊴ … ⊴ Pm = P is a sequence of subgroups all normal in P, such that for each 1 ≤ i ≤ m,α|Pi ≡ IdPi (mod Pi-1). Thenα = IdP.

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 = IdP.