Finite groups in which every maximal invariant subgroup of order divisible by $p$ is nilpotent

TL;DR

This paper addresses the structure of finite groups $G$ with a coprime action by $A$ for a fixed prime $p$ dividing $|G|$, under the condition that every maximal $A$-invariant subgroup of order divisible by $p$ is nilpotent. It proves a complete characterization: such $G$ is either nilpotent or has one of four explicit configurations built from normal Sylow subgroups and $A$-invariant subgroups with precise normality and commutativity properties (cases (2)–(4)). The approach blends a reduction to solvable groups via Beltrán–Shao-type results with a detailed case analysis of semidirect/product decompositions, identifying unique maximal nilpotent $A$-invariant subgroups ($P_0$, $Q_0$, $R_0$) that govern the global structure. This work extends Schmidt-type classifications under coprime actions and sharpens understanding of when maximal $A$-invariant subgroups enforce nilpotency, impacting the taxonomy of finite groups under constrained automorphism actions.

Abstract

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms. For any fixed prime divisor $p$ of $|G|$, we provide a complete characterization of the structure of a group $G$ in which every maximal $A$-invariant subgroup of order divisible by $p$ is nilpotent.

Theorems & Definitions (7)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Corollary 1.6
• proof