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∈ G\ { 1 } with G= NG(⟨ x⟩) NG(⟨ y⟩) , where we are denoting by NG(⟨ x⟩) and by NG(⟨ y⟩) the normalizers of the cyclic subgroups ⟨ x⟩ and ⟨ y⟩ . As a consequence of this result, we classify all possible group elements x, y∈ G\ { 1 } with G= CG(x) CG(y) .