As we know, a finite group $G$ is a rational group if $\chi (g)\in\mathbb{Q}$, where $\chi$ is every irreducible charahter and $g\in G$. I have an interesting question that is "Is 2-Sylow subgroup of a rational group also a rational group?"


Any hints will be appreciated :)



Is 2-sylow subgroup of a rational group also a rational group?
This had been a long standing conjecture, but it has now been answered negatively. Isaacs and Navarro have found a counterexample. 


