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

TL;DR

This paper characterizes the structure of finite groups with a coprime automorphism action, where all maximal invariant subgroups of order divisible by a prime p are nilpotent, extending understanding of group invariants under automorphisms.

Contribution

It provides a complete characterization of finite groups with coprime automorphisms where maximal invariant subgroups of order divisible by p are nilpotent, a novel structural insight.

Findings

Complete classification of such groups

Conditions for nilpotency of invariant subgroups

Implications for automorphism group 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.