Groups in which all characters are rational. 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?
 A: 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 $\left| G \right|$.
A: All Weyl groups have this property. So as for families, the hyperoctahedral groups (signed permutations), and their index 2 subgroups (elements defined by having an even number of sign changes).
A: All groups that be constructed from symmetric groups via cross products and wreath products have this property.  See Section 3 of my paper "Mass formulas for local Galois representations to wreath products and cross products" http://arxiv.org/pdf/0804.4679v1.pdf
So, for example, $((S_7 \wr S_4) × S_3) \wr S_8$ has a rational character table.  In fact, taking cross products and wreathing with $S_n$ preserves the property you are asking about (see above reference).
This includes several of the examples given: $(\mathbb Z/2\mathbb Z)^n,$  hyperoctahedral groups, and Sylow 2-subgroups of $S_n$.  I am not sure if the index 2 subgroups of hyperoctahedral groups can be constructed from symmetric groups via cross products and wreath products.
A: Sylow $2$-subgroups of the symmetric group $S_n$ of degree $n$ are rational.
There was a longstanding conjecture on rational groups saying that  Sylow $2$-subgroups of a rational group are also rational. This has been refuted by I. M. Isaacs and G. Navarro
in [Sylow 2-subgroups of rational solvable groups, Mathematische Zeitschrift,
December 2012, Volume 272, Issue 3-4, pp 937-945.]
