(with Bob Guralnick)

**Publication status:** appeared in: ``Groups,
Rings and Algebras: Proceedings of a Conference in Honor of Donald S. Passman'',
Contemporary Mathematics **420**, 141-162 (2006);

arXiv:
math.GR/0511191

**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.

**Electronic preprint:**

dvi
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ps
377KB |
pdf
218KB |