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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 ⊆ GL(n,Q). This note contains strengthened versions of these results which in particular show that a 2-subgroup of GL(n,Q) of maximum possible order contains a reflection. §1. Statements of Results If ` is a natural number, let Q(2`) denote the cyclotomic field which contains exactly 2` roots of 1, thus Q(2) = Q. For natural numbers ` and m, let M(m, `) denote the group of all monomial matrices whose non-zero entries are `th roots of 1. Then |M(m, `)| = `(m!). Let p be a prime. For a natural number u let up denote the p-part of u. If `p = p , let P (m, p) be a Sylow p-group of M(m, `). As |M(m, `)|p = p(m!)p = |M(m, p)|p, it may be assumed that P (m, p) ⊆M(m, p). Furthermore, |P (m, pk)| = pNp(m,pk), with Np(m, p ) = mk + ∑

Finite linear groups and theorems of Minkowski and Schur