On finite groups in which every non-nilpotent maximal invariant subgroup is normal

Abstract

Let [Formula: see text] and [Formula: see text] be finite groups such that [Formula: see text] acts coprimely on [Formula: see text] by automorphisms, we prove that if every non-nilpotent maximal [Formula: see text]-invariant subgroup of [Formula: see text] is normal, then [Formula: see text] is [Formula: see text]-nilpotent or [Formula: see text]-closed for each prime divisor [Formula: see text] of [Formula: see text], which improves a theorem obtained by Beltán and Shao.