Prime divisors of conjugacy class lengths in finite groups (Q1175684)

Let \(G\) be a finite group. Let \(\rho'(G)\) denote the set of all prime divisors of \(| G/Z(G)|\), \(\pi(G)\) the set of all prime divisors of \(| G|\), and \(\Delta(G)\) the set of all primes \(p\in \pi(G)\) such that \(G\) is not \(p\)-nilpotent with abelian Sylow \(p\)-subgroups. Let \(G_ p\) be a Sylow \(p\)-subgroup of \(G\) and \(n_ p(G)=| N_ g(G_ p): C_ G(G_ p)|\). The author proves the following theorem: Let \(G\) be a non-abelian group. Then \(\sigma'(G)>\sum_{p\in \Delta(G)}(n_ p-1)/n_ p\).