Groups with the given set of the lengths of conjugacy classes. (Q2813419)

Let \(G\) be a finite group and let \(\mathrm{cs}(G)\) be the set of the lengths of the conjugacy classes of \(G\). This paper deals with the problem of determining the structure of groups \(G\) such that \(\mathrm{cs}(G)=\{1,p^{e_1}n_1,p^{e_2}n_2,\ldots,p^{e_k}n_k\}\) with \(p\) a prime number, \(n_1>n_2>\cdots>n_k\) positive integers coprime to \(p\) and \(1=e_1<e_2<\cdots<e_k\). The main result proved by the author is: If \(G\) satisfies the above conditions and \(|\mathrm{cs}(G)|=4\), then \(p=2\) and \(G/Z(G)\simeq\mathrm{PGL}_2(q^n)\) with \(q\) an odd prime.