Rational group (reviewed)

A group $G$ is rational if all of its characters are rational valued. (Equivalently every $g$ and $g^k$ are conjugate for $k$ coprime to $|G|$.) It is an unsolved problem to determine what cyclic groups of order $p$ can be composition factors of rational groups.

Rational groups are sometimes also known as Q-groups, though some authors reserve that term for a stronger property that every representation of G can be realized over $\mathbb{Q}$. (According to this terminology, $Q_8$ is rational but not a Q-group.)