On the number of conjugacy classes of a finite solvable group. II.
[For part I see Bull. Lond. Math. Soc. 32, No. 6, 668-672 (2000; Zbl 1024.20016).] This paper proves the following theorem: Let \(G\) be a finite soluble group and let \(p\) be a prime where \(p^2\) divides the order of \(G\). Then \(G\) has at least \((49p+1)/60\) conjugacy classes. Since we know that the number of conjugacy classes is the same as the number of irreducible characters, the same result applies to the number of irreducible characters. The authors use this fact and really prove it in that form. A key result is the observation that the Alperin-McKay conjecture holds for soluble groups [\textit{T. Okuyama} and \textit{M. Wajima}, Osaka J. Math. 17, 801-806 (1980; Zbl 0446.20003)]. This enables the authors to use induction and show that they need only consider the case where the Sylow \(p\)-subgroup is a normal elementary Abelian subgroup of \(G\) and \(O_{p'}(G)=1\).
scientific article; zbMATH DE number 2037081