On sets of matrices with coefficients in a division ring

Abstract

A number of recent books deal with the theory of groups of linear transformations and its connection with the theory of algebras(1). Most of the work has been restricted to the case of completely reducible systems or, in other words, to semisimple algebras. There are, however, a number of questions which make it desirable not to neglect the other case. The aim of this and a following paper is a study of such not completely reducible systems, in particular of their regular representations. It appeared necessary to start again right from the beginning of the theory, in order to add a number of remarks to well known results and methods(2). The coefficients of the matrices in this paper are taken from an arbitrary division ring K (=skew field or noncommutative field K). This is a generalization of the ordinary theory which does not always work smoothly. For instance, the (left) rank of a ring of matrices t is not invariant under similarity transformation. This implies that similar rings 2L and ?1, may have different regular representations. Yet it is possible to derive a number of results which, in the case of a commutative K, imply the fundamental theorems of Frobenius, Burnside, Loewy, I. Schur and Wedderburn. Sections 1 and 2 deal with a number of group-theoretical remarks. The first of these are concerned with the Jordan-Holder theorem. The connection between two composition series is studied more closely, and it is proved that sets of residue systems can be chosen such that they can be used in either composition series. Further, the upper and lower Loewy series of a group are studied. It is shown that the ith factor groups in both have a common constituent. This implies the theorem of Krull and Ore(3) that both series have the same length. In Section 3, the necessary tools from the theory of matrices are described briefly. The following two sections contain an application of the group-theoretical methods to the study of the irreducible and the Loewy constituents of a set of matrices. In Section 6, a number of further remarks are added, for instance a generalization of a theorem of A. H. Clifford(').