The automorphism group of finite p-groups

Abstract

It is well known [2; Theorem 12.2.21 that if G is a finite p-group with Frattini subgroup @, then the group 5 of automorphisms of G fixing G/Q elementwise is a finite p-group and therefore nilpotent. In this paper we shall obtain an upper bound for the nilpotency class of 3 (Theorem 3). This is done by relating a certain central series of G to a central series of 5. The argument uses induction (Theorem 2 giving information about the center of 3) and depends on a partly well-known general result (Theorem 1 and Corollary 1) showing how certain subgroups of a group lead to normal subgroups of its group of automorphisms. We conclude by applying the theory to Abelian p-groups, wreath products, etc.