A note on character kernels in finite groups of prime power order. (Q2481712)

Let \(p\) be a prime and \(G\) a finite nonabelian \(p\)-group. In the paper under review it is proved that the following conditions are equivalent: (i) The kernels of the nonlinear irreducible characters of \(G\) form a chain (with respect to inclusion); (ii) if \(N\) is a proper subgroup of \(G'\) that is normal in \(G\), then \(N\) is the kernel of an irreducible character of \(G\); and (iii) \(G\) is of maximal class, or \(G'\) is the unique minimal normal subgroup of \(G\). The result is a partial answer to a question in a forthcoming book of \textit{Y. Berkovich} [Groups of prime power order. I].