A dual version of Huppert's conjecture on conjugacy class sizes. (Q486457)

Let \(G\) be a finite group and \(\text{cs}(G)\) be the set of the sizes of conjugate classes of \(G\). The well-known conjecture of J. G. Thompson (posed in 1988 in a communication to W. Shi), which is Problem 12.38 in the Kourovka notebook [\textit{V. D. Mazurov} (ed.) and \textit{E. I. Khukhro} (ed.), The Kourovka notebook. Unsolved problems in group theory. 17th ed. Novosibirsk: Institute of Mathematics, Russian Academy of Sciences, Siberian Div. (2010; Zbl 1211.20001)], states the following: If \(S\) is a non-abelian finite simple group and \(G\) is a finite group such that \(Z(G)=1\) and \(\text{cs}(G)=\text{cs}(S)\), then \(G\cong S\). \textit{N. Ahanjideh} [J. Algebra 344, No. 1, 205-228 (2011; Zbl 1247.20015)] proved Thompson's conjecture for \(\text{PSL}_n(q)\). In this article, the authors improve this result for \(\text{PSL}_2(q)\). They prove that if \(\text{cs}(G)=\text{cs}(\text{PSL}_2(q))\) for \(q>3\), then \(G\cong\text{PSL}_2(q)\times A\), where \(A\) is abelian. Their proof does not depend on the classification of finite simple groups.