Return to Revisions

Here's one characterization that I learned from Serre (see Definition 7.1.1 in his Topics in Galois Theory (p.65)): an element $g$ of a finite group $G$ satisfies $\chi(g) \in {\bf Q}$ for all characters $\chi$ iff $g$ is conjugate in $G$ to $g^m$ for all $m$ relatively prime to the exponent $e(g)$. [If $m$ is not coprime to $e(g)$ then $e(g^m) \lt e(g)$ so $g^m$ cannot possibly be conjugate to $g$.] It is enough to check this for all $m$ relatively prime to $\left| G \right|$. In particular, all character values are rational iff every group element is conjugate to its $m$-th power for all $m$ coprime to $\left| G \right|$.