Orders of Finite Groups of Matrices

TL;DR

This paper offers a new, accessible proof of Schur's theorem on the least common multiple of orders of finite matrix groups over number fields, extending classical results with a focus on algebraic structures.

Contribution

It provides a novel proof of Schur's theorem using Minkowski's method, generalizing to arbitrary number fields and making the proof more accessible.

Findings

New proof of Schur's theorem on matrix group orders

Extension of results to arbitrary number fields

Accessible exposition for graduate students

Abstract

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.