Action of Aut(G) on the set of maximal subgroups of p-groups

Abstract

In this paper, we mainly study the action of Aut(G) on the set ω of all maximal subgroups of G, and we use P to denote the action of Aut(G) on ω. When G is a finite non-cyclic abelian p-group, P is transitive. When G is minimal non-abelian, there are three cases. If G is quaternion group, P is transitive; if G is a metacyclic group, P is non-transitive; if G is not a metacyclic group, P is transitive.