Linear Groups over Division Rings

Abstract

Let R be a division ring and V a finite dimensional vector space over R. The concern of this chapter is the investigation of the group GL(V) and its important subgroups. In an initial section it is proved that GL(V) is generated by transvections and dilations in a certain efficient way. Also, the orders of the linear groups are computed when R is finite. A subsequent section considers the subgroup E(V) generated by the transvections alone. This group is equal to the group EX(V), already studied in §1.2C, for any basis X of V. In addition, E(V) is the kernel of the Dieudonne determinant and the quotient E(V)/Cen E(V) is a simple group, aside from the two exceptions where dim V = 2 and R is the Galois field \( {\mathbb{F}_2}{\text{ or }}{\mathbb{F}_3} \). For a division ring R which is finite dimensional over its center this section also introduces the “norm one” subgroup of GL(V). This group is denoted SL(V). If R is a field SL(V) is the kernel of the determinant so that this notation is consistent with that already in use. The group SL(V) always contains E(V). The question as to whether it is equal to E(V) in general was for years an important open problem known in the literature as the problem of “Tanaka-Artin” or (in a more general setting) as the problem of “Kneser-Tits”. This question, and its more recent negative solution, is discussed without proofs. A final section develops a Bruhat decomposition for GL(V). This in combination with the K-Theory of Chapter 1 leads to several presentation theorems for the linear groups. The chapter concludes with a discussion of the Theorems of Matsumoto and Merkurjev-Suslin.