A direct proof of Müller’s result on automorphisms of finite p-groups

Abstract

Müller proved that, if G is a finite p-group which is neither elementary abelian nor extra-special, then ${{\,\mathrm{Aut}\,}}^{\Phi }(G) / {{\,\mathrm{Inn}\,}}(G)$ is a non-trivial normal p-subgroup of the group of outer automorphisms of G. Here ${{\,\mathrm{Aut}\,}}^{\Phi }(G)$ denotes the group of all automorphisms of G that centralize the Frattini quotient $G/\Phi (G)$ of G. We give a new direct proof, which avoids, in particular, the use of cohomological considerations.