Solvability of finite groups with four conjugacy class sizes of certain elements. (Q2922938)

This paper makes some new contributions to the study of groups with exactly four conjugacy class sizes of certain elements. First, they correct a mistake in a paper by \textit{Q. Kong} and \textit{Q. Liu} [Bull. Aust. Math. Soc. 88, No. 2, 297-300 (2013; Zbl 1281.20029); erratum 9, No. 3, 522-523 (2014; Zbl 1290.20017)]. They also generalize a result of \textit{A. Beltrán} and \textit{M. J. Felipe} [J. Algebra Appl. 11, No. 2, 1250036 (2012; Zbl 1251.20031); Corrigendum 11, No. 6, Paper No. 1292001 (2012; Zbl 1255.20032)] by proving the following. Theorem. Let \(G\) be a finite group whose conjugacy class sizes of primary and biprimary elements are 1, \(m\), \(n\) and \(mn\), where \(m\) and \(n\) are positive integers greater than 1 such that \(m\) does not divide \(n\), and \(n\) does not divide \(m\). Then \(G\) is solvable and there exist two distinct primes \(p\) and \(q\) such that up to central factors, \(G\) is a \(\{p,q\}\)-group.