THE FINITE IRREDUCIBLE LINEAR 2-GROUPS OF DEGREE-4 - INTRODUCTION

We present, a classification of the finite irreducible 2-subgroups of GL(4, C); that is, we give a parametrised list of representatives for the conjugacy classes of such groups. Each group listed is defined by a generating set of monomial matrices. There are essentially three pos sibilities for the projection of an irreducible monomial 2-group into the group of all permutation matrices. The classification problem acco rdingly falls into three separate cases. Each case may be handled by a general method consisting of three major steps. Techniques for applyi ng the method to the most difficult case are developed in detail, so t hat the other two cases may then be dealt with routinely. The techniqu es used include elementary character theory, a method for drawing the Hasse diagram of the submodule lattice of a direct sum, and cohomology theory, particularly the calculation of 2-cohomology by means of the Lyndon-Hochschild-Serre spectral sequence. Related questions concernin g isomorphism between the listed groups, and Schur indices over Q, are also considered.