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

TL;DR

The paper extends solvability criteria for finite groups under coprime action by focusing on maximal $A$-invariant subgroups that are direct products of isomorphic simple groups. It proves that if a nontrivial $A$-invariant normal subgroup $N\le M$ exists and every non-nilpotent maximal $A$-invariant subgroup not containing $N$ has index a prime-power, with the exclusion that $PSL_2(7)$ is a composition factor, then $G$ is solvable, using a minimal counterexample argument and detailed analysis of normalizers and subgroup structure. The approach combines coprime-action lemmas with a case analysis on the structure of maximal subgroups to rule out non-solvable configurations. The results generalize and integrate prior findings by Shi, Beltrán, Shao, and Guralnick, clarifying the role of index constraints and the absence of $PSL_2(7)$ in ensuring solvability under coprime action. Overall, the work advances understanding of how invariant direct-product structures interact with maximal-subgroup indices to determine solvability in finite groups.

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.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• Lemma 2.1
• Lemma 2.2