On finite groups in which some maximal invariant subgroups have indices a prime or the square of a prime

TL;DR

This paper investigates the structure of finite groups with certain maximal invariant subgroups of prime or prime-square index under coprime automorphism actions, extending classical theorems and providing a full characterization.

Contribution

It generalizes Hall's theorem to broader contexts and characterizes finite groups with specific invariant subgroup index conditions under automorphisms.

Findings

Generalized Hall's theorem for groups with invariant subgroups of prime or prime-square index.

Characterized finite groups where non-nilpotent maximal invariant subgroups have prime index.

Provided a complete classification of groups with certain invariant subgroup properties.

Abstract

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms, we first prove some results on the solvability of finite groups in which some maximal $A$-invariant subgroups have indices a prime or the square of a prime. Our results generalize Hall's theorem and some other known results. Moreover, we obtain a complete characterization of finite groups in which every non-nilpotent maximal $A$-invariant subgroup that contains the normalizer of some $A$-invariant Sylow subgroup has index a prime.