On an extension of Hall’s criterion

Abstract

Let A and G be finite groups such that A acts coprimely on G by automorphisms, assume that G has a maximal A-invariant subgroup M  that is a direct product of some isomorphic simple groups, we prove that if G has a nontrivial A-invariant normal subgroup N such that N  ≤ M and every non-nilpotent maximal A-invariant subgroup K of G not containing N has index a prime or the square of a prime, then G is  solvable.