  We present a new proof of a theorem of Schur's determining the least common
multiple of the orders of all finite groups of complex $n \times n$-matrices
whose elements have traces in the field of rational numbers. The basic method
of proof goes back to Minkowski and proceeds by reduction to the case of finite
fields. For the most part, we work over an arbitrary number field rather than
the rationals. The first half of the article is expository and is intended to
be accessible to graduate students and advanced undergraduates. It gives a
self-contained treatment, following Schur, over the field of rational numbers.
