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  • Print publication year: 2011
  • Online publication date: September 2011

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 PQ, 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 < KNG(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 = P0P1 ⊴ … ⊴ Pm = P is a sequence of subgroups all normal in P, such that for each 1 ≤ im,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 qp. In this case, for each gP, 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.