Finite groups in which some particular non-nilpotent maximal invariant subgroups have indices a prime-power

TL;DR

This paper investigates the structure of finite groups with specific invariant subgroups, showing under certain conditions that such groups must be solvable, especially when certain non-nilpotent maximal invariant subgroups have prime-power indices.

Contribution

It establishes new conditions under which finite groups with particular invariant subgroups are guaranteed to be solvable, extending understanding of group structure under automorphism actions.

Findings

Groups with specified invariant subgroups are solvable under given conditions.

Non-nilpotent maximal invariant subgroups with prime-power indices influence group solvability.

The absence of $PSL_2(7)$ as a composition factor is crucial for the main result.

Abstract

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms, assume that $G$ has a maximal $A$-invariant subgroup $M$ that is a direct product of some isomorphic simple groups, we prove that if $G$ has a non-trivial $A$-invariant normal subgroup $N$ such that $N\leq M$ and every non-nilpotent maximal $A$-invariant subgroup $K$ of $G$ not containing $N$ has index a prime-power and the projective special linear group $PSL_2(7)$ is not a composition factor of $G$, then $G$ is solvable.