The characters of a group of type \(X\wr\mathbb{Z}_p\). (Q2577172)

Given a prime \(p\) and a finite group \(X\) with known character table, the author calculates the irreducible complex characters of the wreath product \(X\wr\mathbb{Z}_p\). Since the Sylow \(p\)-subgroups of the symmetric groups can be constructed from groups isomorphic to \(\mathbb{Z}_p\) by means of direct and wreath products, the author obtains a simple inductive method of finding the character tables for these groups. In particular, he proves that irreducible complex characters of the Sylow \(2\)-groups of the symmetric groups possess only rational values, which solves Problem 15.25 of The Kourovka notebook [\textit{V.~D.~Mazurov} and \textit{E.~I.~Khukhro} (eds.), The Kourovka notebook. Unsolved problems in group theory. Novosibirsk (2002; Zbl 0999.20001)].