A generalization of Szep's conjecture for almost simple groups

Gill, Nick; Giudici, Michael and Spiga, Pablo (2023). A generalization of Szep's conjecture for almost simple groups. Vietnam Journal of Mathematics (Early access).

DOI: https://doi.org/10.1007/s10013-023-00635-1

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 $G=C_G(x)C_G(y)$.

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