The Symmetric groups $S\_n$ has interesting property that all complex irreducible characters are rational (i.e. $\chi(g)\in \mathbb{Q}$ for all $\mathbb{C}$-irreducible characters $\chi$,$\forall g\in S\_n$). 


**Question:** What are other families of (finite) groups where all complex irreducible characters are rational? Are such (finite) groups characterised?



Groups in which all characters are rational.
Here's one characterization that I learned from Serre
(see Definition 7.1.1 in his
[*Topics
in Galois Theory*](http://www.math.mcgill.ca/~darmon/pub/Articles/Serre/c.pdf) (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 $\left| G \right|$.


