Czechoslovak Mathematical Journal, online first, 10 pp.

Thompson's conjecture for the alternating group of degree $2p$ and $2p+1$

Azam Babai, Ali Mahmoudifar

Received July 25, 2016.   First published October 5, 2017.

Azam Babai, Department of Mathematics, University of Qom, Alghadir Blvd., 37185-3766 Qom, Iran, e-mail: a_babai@aut.ac.ir; Ali Mahmoudifar, Department of Mathematics, Tehran-North Branch, Islamic Azad University, South Makran Street, 1651153311 Tehran, Iran, e-mail: a_mahmoodifar@iau-tnb.ac.ir

Abstract: For a finite group $G$ denote by $N(G)$ the set of conjugacy class sizes of $G$. In 1980s, J. G. Thompson posed the following conjecture: If $L$ is a finite nonabelian simple group, $G$ is a finite group with trivial center and $N(G) = N(L)$, then $G\cong L$. We prove this conjecture for an infinite class of simple groups. Let $p$ be an odd prime. We show that every finite group $G$ with the property $Z(G)=1$ and $N(G) = N(A_i)$ is necessarily isomorphic to $A_i$, where $i\in\{2p,2p+1\}$.

Keywords: finite group; conjugacy class size; simple group

Classification (MSC 2010): 20D05, 20D60

DOI: 10.21136/CMJ.2017.0396-16

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