Normal sections, class sizes and solvability of finite groups. (Q397848)

That the sizes of conjugacy classes of a finite group have an influence on the structure of the group has long been investigated. In particular, that a finite group with exactly 3 conjugacy class sizes is soluble was proved by \textit{N. Ito} [Osaka J. Math. 7, 231-251 (1970; Zbl 0198.04305)].NEWLINENEWLINE In this article the authors generalise this result in the following manner. Suppose \(N\) is a normal subgroup of the finite group \(G\) and consider the \(G\)-conjugacy class sizes of elements in \(N\). Then, if there are exactly 3 such conjugacy class sizes, \(N\) is soluble. As the authors comment, in general the \(G\)-class sizes of \(N\) give little information on the class sizes of \(N\), which makes the result more surprising. The proof is technical and uses the Classification of Finite Simple Groups.