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).