Finite groups in which every maximal subgroup is nilpotent or normal or has $p'$-order

Abstract

Let G be a finite group and p a fixed prime divisor of |G|. Combining the nilpotence, the normality and the order of groups together, we prove that if every maximal subgroup of G is nilpotent or normal or has p ′ -order, then (1) G is solvable;(2) G has a Sylow tower; (3) There exists at most one prime divisor q of |G| such that G is neither q-nilpotent nor q-closed, where q = p.