Finite linear groups and theorems of Minkowski and Schur

Abstract:

Let $G$ be a finite group with a faithful rational valued character of degree $n$. A theorem of I. Schur gives a bound for the order of $G$ in terms of $n$, generalizing an earlier result of H. Minkowski who showed that the same bound holds if $G\subseteq GL(n,\mathbf {Q})$. This note contains strengthened versions of these results which in particular show that a $2$-subgroup of $GL(n,\mathbf {Q})$ of maximum possible order contains a reflection.