Powers of irreducible characters and conjugacy classes in finite groups. (Q2878802)

Let \(G\) be a finite group and \(G'\) its commutator subgroup. The derived covering number \(\text{dcn}(G)\) is defined as the smallest positive integer \(n\) such that \(C^n=G'\) for all non-central conjugacy classes \(C\) of \(G\). If such an integer exists, then \(\text{dcn}(G)\) is said to be finite, otherwise infinite.NEWLINENEWLINE The authors are interested in conditions guaranteeing that \(\text{dcn}(G)\) is finite and in obtaining upper bounds in this case, and they prove some results in these directions. For example, they show that if \(G\) is nonabelian and \(G'\) is a minimal normal subgroup, then \(\text{dcn}(G)\) is finite. Also, if \(G\) is nonabelian and \(\text{dcn}(G)\) is finite, then one of the following holds: (i) \(G'\) is a minimal normal subgroup in \(G\), (ii) \(G\) is nilpotent of class 2, or (iii) \(G'\) is perfect and \(G'/Z(G')\) is minimal normal in \(G/Z(G')\), and \(Z(G')=G'\cap Z(G)\). Moreover, if \(G\) is a real finite group and \(G'\) is a nonabelian simple subgroup, then \(\text{dcn}(G)\leq 2(k(G)-1)\), where \(k(G)\) is the number of conjugacy classes of \(G\). -- Similarly the \textit{derived character covering number} is introduced to study the analogous questions for powers of characters.