A generalization of Szep's conjecture for almost simple 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 $G={\mathbf{N}}_G(⟨x⟩){\mathbf{N}}_G(⟨y⟩)$, where we are denoting by ${\mathbf{N}}_G(⟨x⟩)$ and by ${\mathbf{N}}_G(⟨y⟩)$ the normalizers of the cyclic subgroups $⟨x⟩$ and $⟨y⟩$. As a consequence of this result, we classify all possible group elements $x,y\in G\setminus \{1\}$ with $G={\mathbf{C}}_G(x){\mathbf{C}}_G(y)$.

36 pages