A group is said to be a $Q$-group if the character of any complex representation is rational valued. A well-known internal characterization of $Q$-groups is the following:



> 
> $G$ is a $Q$-group if and only if elements which generate the same cyclic subgroup of $G$ lie in the same conjugacy class.
> 
> 
> 


See for example: Corollary 2 to theorem 29 in Serre's *Linear Representations of Finite Groups* as suggested in <https://mathoverflow.net/a/10652/9672>, or Kletzing's *Structure and Representations of $Q$-groups*.


I would be grateful if someone could tell me where this theorem first appears (to whom is it due).



To whom is the internal characterization of $Q$-groups due?
It's hard to pin down who first made this remark because it is a corollary of the very basic fact that if $\sigma$ is an automorphism of the cyclotomic field $Q\_{|G|}$, then there exists an integer $m$ coprime to $|G|$ such that for every character $\chi$ of $G$, we have $\chi(x)^\sigma = \chi(x^m)$. The earliest explicit reference I can find is in Huppert's book *Endliche Gruppen I* in 1967. It appears there as Satz V 13.7 (b), on page 537.


