Lengths of conjugacy classes of finite solvable groups. II
[This review concerns also the preceding item Zbl 0798.20013.] Denote by \(w(n)\) the number of distinct prime divisors of the positive integer \(n\); if \(G\) is a finite group, set \(\alpha(G):=\max\{w(| G:C_ G(x)|)\mid x\in G\}\) and \(\beta(G):=\{p\in\pi(G)\mid p|| G:C_ G(x)|\) for some \(x\in G\}\). The most important general result of the papers is that if \(G\) is a solvable group, then \(|\beta(G)|\leq 4\alpha (G)+6\). Several other better estimates are obtained if additional constraints are imposed on \(G\); for example, if \((| G|,6)=1\), then \(|\beta(G)|\leq 3.25\alpha(G) +6\). If \(G/R(G)\) is cyclic, where \(R(G)\) is the intersection of the maximal normal subgroups of \(G\), the former estimates can be improved further to \(|\beta(G)|\leq 3\alpha(G)+6\), or \(|\beta(G)|\leq 3\alpha(G)\) if, moreover, \((| G|,6)=1\).
scientific article; zbMATH DE number 130514