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

Jiangtao Shi, Yunfeng Tian

TL;DR

This work studies finite groups $G$ with a coprime action by a group $A$ on $G$, focusing on when maximal $A$-invariant subgroups have index equal to a prime or the square of a prime. It generalizes Hall's theorem and prior coprime-action results by proving that if every non-nilpotent maximal $A$-invariant subgroup containing the normalizer of some $A$-invariant Sylow subgroup has index a prime or $p^2$, then $G$ is solvable; it also derives a Sylow-tower structure under these hypotheses. The paper further provides a complete characterization (Theorem 1.7) showing that, in particular, the largest prime divisor $p$ of $|G|$ forces either an $A$-invariant Sylow $p$-subgroup to be normal or the existence of a normal $p$-complement, and yields a Sylow-tower description. These results extend Hall-type solvability criteria and clarify the Sylow-structure consequences of restricted index conditions under coprime actions.

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.

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

TL;DR

This work studies finite groups with a coprime action by a group on , focusing on when maximal -invariant subgroups have index equal to a prime or the square of a prime. It generalizes Hall's theorem and prior coprime-action results by proving that if every non-nilpotent maximal -invariant subgroup containing the normalizer of some -invariant Sylow subgroup has index a prime or , then is solvable; it also derives a Sylow-tower structure under these hypotheses. The paper further provides a complete characterization (Theorem 1.7) showing that, in particular, the largest prime divisor of forces either an -invariant Sylow -subgroup to be normal or the existence of a normal -complement, and yields a Sylow-tower description. These results extend Hall-type solvability criteria and clarify the Sylow-structure consequences of restricted index conditions under coprime actions.

Abstract

Let and be finite groups such that acts coprimely on by automorphisms, we first prove some results on the solvability of finite groups in which some maximal -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 -invariant subgroup that contains the normalizer of some -invariant Sylow subgroup has index a prime.
Paper Structure (4 sections, 9 theorems)

This paper contains 4 sections, 9 theorems.

Table of Contents

  1. Introduction
  2. Two Necessary Lemmas
  3. Proof of Theorem 1.2
  4. Proof of Theorem 1.7

Key Result

Theorem 1.1

[5, Theorem A] Let $G$ and $A$ be groups of coprime orders and assume that $A$ acts on $G$ by automorphisms. If the index of every maximal $A$-invariant subgroup of $G$ is prime or the square of a prime, then $G$ is solvable.

Theorems & Definitions (9)

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