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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).

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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.

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  • $\begingroup$ I was hoping that someone would point me to a paper of Frobenius. $\endgroup$ – Amritanshu Prasad Feb 6 '14 at 6:47

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