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).


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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