On the solvability of groups with four class sizes. (Q2885397)

Finite groups whose set of irreducible complex character degrees is 1, \(m\), \(n\) and \(mn\) for positive integers \(m\) and \(n\) are known to be solvable [\textit{G. Qian} and \textit{W. Shi}, J. Group Theory 7, No. 2, 187-196 (2004; Zbl 1070.20010)]. In the paper under review the authors prove a partial analogue for conjugacy class sizes in place of character degrees. More precisely, let \(G\) be a finite group whose conjugacy class sizes are 1, \(m\), \(n\) and \(mn\), where \(m\) and \(n\) are positive integers greater than 1 such that \(m\) does not divide \(n\), and \(m\) does not divide \(n\). Then there exist two distinct primes \(p\) and \(q\) such that up to central factors, \(G\) is a \(\{p,q\}\)-group.