Conjugacy class lengths of metanilpotent groups (Q1356826)

For a finite group \(G\), let \(\pi(G)\) denote the set of prime divisors of \(|G|\); if \(g\in G\), let \(\sigma(g)\) be the set of prime divisors of \(|G:C_G(g)|\), let \(\sigma^*(G)=\max\{|\sigma(g)|:g\in G\}\) and let \(\rho^*(G)=\pi(G/Z(G))\). B. Huppert has asked whether the inequality \(|\rho^*(G)|\leq 2\sigma^*(G)\) holds for soluble groups \(G\). The authors prove that if \(G\) is a finite metanilpotent group and if \(p\) is the smallest prime divisor of \(|G:G'Z(G)|\), then \(|\rho^*(G)|<((2p-1)/(p-1))\sigma^*(G)\). This inequality is then shown to be best possible by letting \(p=2\) and constructing an infinite family of supersoluble metabelian groups \(G_n\) satisfying \(\lim_{n\to\infty}|\rho^*(G_n)|/\sigma^*(G_n)=3\). Thus Huppert's question is solved in the negative, in spite of a number of particular cases in which it is known to hold (it was verified for \(\sigma^*(G)=1\) by \textit{D. Chillag} and \textit{M. Herzog} [J. Algebra 131, No. 1, 110-125 (1990; Zbl 0694.20015)], for \(\sigma^*(G)=2\) by \textit{P. A. Ferguson} [J. Algebra 143, No. 1, 25-28 (1991; Zbl 0735.20011)] and for \(\sigma^*(G)=3\) by \textit{C. Casolo} [Manuscr. Math. 82, No. 2, 171-189 (1994; Zbl 0819.20024)]).