On the units in a character ring (Q1185401)

Denote the algebraic closure of the rational field \(Q\) in the complex number field \(C\) by \(\bar Q\) and the ring of algebraic integers in \(\bar Q\) by \(\bar Z\). Moreover, let \(R(G)\) be a character ring of \(G\). The author proves that if \(G\) is a finite group then any unit of finite order in \(\bar Z\otimes R(G)\) has the form \(\varepsilon \chi\) for some linear character \(\chi\) and some unit \(\varepsilon\) in \(\bar Z\), and he applies this result to conclude that if \(G\) and \(G'\) are finite groups such that \(R(G)\cong R(G')\) as rings, then \(G/D(G)\cong G'/D(G')\), where \(D(G)\) and \(D(G')\) are the commutator subgroups of \(G\) and \(G'\) respectively.