A generalization of Szep's conjecture for almost simple groups

TL;DR

This paper generalizes Szep's conjecture for almost simple groups by classifying elements based on their normalizers and centralizers, excluding certain orthogonal groups, and provides a comprehensive structural understanding.

Contribution

It introduces a broad classification of elements in almost simple groups related to their normalizers and centralizers, extending Szep's conjecture to new group classes.

Findings

Classified elements with specific normalizer product properties

Identified conditions excluding orthogonal groups with Witt defect zero

Extended Szep's conjecture to a wider class of groups

Abstract

We prove a natural generalization of Szep's conjecture. Given an almost simple group $G$ with socle not isomorphic to an orthogonal group having Witt defect zero, we classify all possible group elements $x,y\in G\setminus\{1\}$ with $G={\bf N}_G (\langle x\rangle){\bf N}_G(\langle y\rangle)$, where we are denoting by ${\bf N}_G(\langle x\rangle)$ and by ${\bf N}_G(\langle y\rangle)$ the normalizers of the cyclic subgroups $\langle x\rangle$ and $\langle y\rangle$. As a consequence of this result, we classify all possible group elements $x,y\in G\setminus\{1\}$ with $G={\bf C}_G(x){\bf C}_G(y)$.