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

TL;DR

This paper characterizes finite groups where every maximal subgroup is either nilpotent, normal, or has order coprime to a fixed prime, showing such groups are solvable with a Sylow tower and specific prime divisibility properties.

Contribution

It establishes new structural conditions on finite groups based on maximal subgroup properties involving nilpotence, normality, and order constraints.

Findings

Groups are solvable under the given maximal subgroup conditions.

Such groups possess a Sylow tower structure.

At most one prime divisor relates to non-q-nilpotent and non-q-closed behavior.

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 $q\neq p$.