Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Finite quasisimple groups of 2×2 matrices over a division ring

B. Hartleya1 and M. A. Shahabi Shojaeia1

a1 Unïversity of Manchester and University of Tabriz, Iran

In 1955 [1], Amitsur determined all the finite groups G that can be embedded in the multiplicative group T* = GL(1, T) of some division ring T of characteristic zero. If G can be so embedded, then the rational span of G in T is a division ring of finite dimension over xs211A, and G acts on it by right multiplication in such a way that every non-trivial element operates fixed point freely. The finite groups admitting such a representation had earlier been determined by Zassenhaus[24; 4, XII. 8], and Amitsur begins by quoting Zassenhaus' results, which show in particular that the only perfect group that can be embedded in the multiplicative group of a division ring of characteristic zero is SL(2,5). The more difficult part of Amitsur's paper is the determination of the possible soluble groups. Here the main tool is Hasse's theory of cyclic algebras over number fields.

(Received May 14 1984)

(Revised October 15 1984)