On the Pronormality of Subgroups of Odd Index in Finite Simple Groups

TL;DR

This paper disproves a conjecture that subgroups of odd index are pronormal in finite simple groups and discusses recent progress in classifying such groups.

Contribution

It provides a counterexample to the conjecture and reviews recent classification results related to pronormal subgroups in finite simple groups.

Findings

Disproved the conjecture that all subgroups of odd index are pronormal in finite simple groups.

Identified specific finite simple groups where subgroups of odd index are pronormal.

Summarized recent progress in classifying finite simple groups with pronormal subgroups of odd index.

Abstract

A subgroup $H$ of a group $G$ is said to be {pronormal} in $G$ if $H$ and $H^g$ are conjugate in $\langle H, H^g \rangle$ for every $g \in G$. Some problems in finite group theory, combinatorics, and permutation group theory were solved in terms of pronormality. In 2012, E. Vdovin and the third author conjectured that the subgroups of odd index are pronormal in finite simple groups. In this paper we disprove their conjecture and discuss a recent progress in the classification of finite simple groups in which the subgroups of odd index are pronormal.