Nilpotent Groups

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Lemma 7.15 Let G be a nilpotent group. Then every maximal subgroupof G is normal in G.□Definition 7.16 The Frattini subgroup Φ(G) ofagroupG is the intersectionof all its maximal subgroups:⋂Φ(G) = M.M maximalin G(If G is an (infinite) group with no maximal subgroups, then Φ(G) =G.)If we apply an automorphism to a maximal subgroup, we map it toanother maximal subgroup. Hence the automorphism group permutes themaximal subgroups of G.Lemma 7.17 If G is a group, then the Frattini subgroup Φ(G) is a characteristicsubgroup of G.□Our final theorem characterising nilpotent finite groups is:Theorem 7.18 Let G be a finite group. The following are equivalent:(i) G is nilpotent;(ii) H
(iv) ⇒ (v): Let P be a Sylow p-subgroup of G and let N = P Φ(G)(which is a subgroup of G, sinceΦ(G) G by Lemma 7.17). Let x ∈ Nand g ∈ G. Thenx −1 x g =[x, g] ∈ G ′ Φ(G) N.Hence x g ∈ N for all x ∈ N and g ∈ G, soN G. Now P is a Sylowp-subgroup of N (since it is the largest possible p-subgroup of G, so iscertainly largest amongst p-subgroups of N). Apply the Frattini Argument(Lemma 6.35):G =N G (P ) N=N G (P ) P Φ(G)=N G (P )Φ(G) (as P N G (P )).From this we deduce that G =N G (P ): for suppose N G (P ) ≠ G. ThenN G (P ) M < G for some maximal subgroup M of G. By definition,Φ(G) M, soN G (P )Φ(G) M
Page 7: Claim: P 1 P 2 ...P j∼ = P1 × P

Nilpotent Groups

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