Publication status: appeared in: ``Groups,
Rings and Algebras: Proceedings of a Conference in Honor of Donald S. Passman'',
Contemporary Mathematics 420, 141-162 (2006);
Abstract: We present a new proof of a theorem of Schur's from 1905 determining the least common multiple of the orders of all finite groups of complex matrices of a given size 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.