Zeinab Akhlaghi, Maryam Khatami, Tung Le, Jamshid Moori, Hung P. Tong-Viet
October 10, 2014

### Abstract

In [J. Algebra 344 (2011), 205–228], a conjecture of J. G. Thompson for PSL n ( q ) was proved. It was shown that every finite group G with the property Z( G ) = 1 and cs( G ) = cs(PSL n ( q )) is isomorphic to PSL n ( q ) where cs( G ) is the set of conjugacy class sizes of G . In this article we improve this result for PSL 2 ( q ). In fact we prove that if cs( G ) = cs(PSL 2 ( q )), for q > 3, then G ≅ PSL 2 ( q ) × A , where A is abelian. Our proof does not depend on the classification of finite simple groups.