A generalization of Szep's conjecture for almost simple groups

Gill, Nick; Giudici, Michael and Spiga, Pablo A generalization of Szep's conjecture for almost simple groups. Vietnam Journal of Mathematics (In Press).

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 $G=N_G(\langle x\rangle)N_G(\langle y\rangle)$ , where we are denoting by $N_G(\langle x\rangle)$ and by $N_G(\langle y\rangle)$ 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 $G=C_G(x)C_G(y)$ .