Finite groups whose maximal subgroups have almost odd index

TL;DR

This work classifies finite groups whose maximal subgroups have almost odd index, unifying invariant-based results with a maximal-subgroup perspective. It delivers a complete almost-simple-group description and a structural reduction for general finite groups via the quotient by $ extbf{O}_{2',2}(G)$, together with new $2$-nilpotency criteria and a second-maximal-subgroup recognition framework for $p$-nilpotency. The results identify explicit exceptional simple factors that can appear in nonsolvable groups (e.g., $A_6$, $A_7$, $PSU_{3}(5)$, $PSL_{2}(q)$ with $q mid 8$, and $G_2(q)$ with $q=5^{2n+1}$) and establish derived-length bounds for Sylow $2$-subgroups. Computational GAP verifications accompany the proofs, enhancing the robustness of the case analyses and enabling reproducibility of the classifications.

Abstract

A recurring theme in finite group theory is understanding how the structure of a finite group is determined by the arithmetic properties of group invariants. There are results in the literature determining the structure of finite groups whose irreducible character degrees, conjugacy class sizes or indices of maximal subgroups are odd. These results have been extended to include those finite groups whose character degrees or conjugacy class sizes are not divisible by $4$. In this paper, we determine the structure of finite groups whose maximal subgroups have index not divisible by $4$. As a consequence, we obtain some new $2$-nilpotency criteria.

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• proof