Field equivalent finite groups. (Q706119)

Following a suggestion of I. M. Isaacs, the authors define two finite groups \(X\) and \(Y\) to be `field equivalent' if there is a bijection \(\chi\to\chi'\) from the set of all irreducible characters of \(X\) to the set of all irreducible characters of \(Y\) such that, for every \(\chi\), the fields of values over the rational numbers of \(\chi\) and of \(\chi'\) are equal, so that \(\mathbb{Q}(\chi)=\mathbb{Q}(\chi')\). The authors prove that if \(G\) is a finite group which is field equivalent to a cyclic group, then \(G\) is a cyclic group. They remark that there exist non-Abelian groups which are field equivalent to elementary Abelian groups, and non-solvable groups which are field equivalent to \(2\)-groups.