Itô's theorem on groups with two class sizes revisited. (Q2891976)

The following is shown: Theorem A. Suppose \(G\) is a finite \(p\)-solvable group and that 1 and \(m\) are the sizes of the conjugacy classes of \(p'\)-elements of prime power order. Then \(m=p^aq^b\) with \(q\) a prime distinct from \(p\); here \(a\geq 0\) and \(b\geq 0\). If \(b=0\), then \(G\) has Abelian \(p\)-complements. If \(b\neq 0\), then \(G=PQ\times A\), with \(P\) and \(Q\) a Sylow \(p\)-subgroup and a Sylow \(q\)-subgroup of \(G\), resp., and \(A\) is contained in the center of \(G\).NEWLINENEWLINE Corollary B. Let \(G\) be a finite group and suppose that \(G\) has exactly two class sizes of elements of prime power order, 1 and \(m\). Then \(m\) is a prime power and \(G\) is nilpotent, where \(G=Q\times A\) with \(Q\) a Sylow \(q\)-subgroup of \(G\) and \(A\) contained in the center of \(G\).NEWLINENEWLINE The paper under review extends work of \textit{R. Baer} [Trans. Am. Math. Soc. 75, 20-47 (1953; Zbl 0051.25702)]; the second and third author of this paper [Bull. Aust. Math. Soc. 67, No. 1, 163-169 (2003; Zbl 1041.20020)]; \textit{N. Itô} [Nagoya Math. J. 6, 17-28 (1953; Zbl 0053.01202)]; \textit{S. Li} [Arch. Math. 67, No. 2, 100-105 (1996; Zbl 0854.20034)] and \textit{X. Zhao} and \textit{X. Guo} [Algebra Colloq. 16, No. 4, 541-548 (2009; Zbl 1187.20017)].