A Generalization of Szep’s Conjecture for Almost Simple Groups

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 $$x,y\in G\setminus \{1\}$$x , y ∈ G \ { 1 } with $$G=\textbf{N}_G(\langle x\rangle )\textbf{N}_G(\langle y\rangle )$$G = N G ( ⟨ x ⟩ ) N G ( ⟨ y ⟩ ) , where we are denoting by $$\textbf{N}_G(\langle x\rangle )$$N G ( ⟨ x ⟩ ) and by $$\textbf{N}_G(\langle y\rangle )$$N G ( ⟨ y ⟩ ) the normalizers of the cyclic subgroups $$\langle x\rangle $$⟨ x ⟩ and $$\langle y\rangle $$⟨ y ⟩ . As a consequence of this result, we classify all possible group elements $$x,y\in G\setminus \{1\}$$x , y ∈ G \ { 1 } with $$G=\textbf{C}_G(x)\textbf{C}_G(y)$$G = C G ( x ) C G ( y ) .