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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