General Linear Groups, Steinberg Groups, and K-Groups

Abstract

This chapter starts with a quick review of some relevant algebra. It then turns to the study of the linear groups over a ring R. The general linear groups, the elementary linear groups, and the linear congruence groups are introduced and their basic properties are established. It is proved for a commutative R and n ≥3 that the elementary group E n (R) is normal in GL n (R). Later the “stable” analogues GL(R) and E(R) are constructed, their normal subgroups are characterized, and the various K1 are defined. Then the linear Steinberg groups St(R) and its important subgroups are developed and it is proved that St(R) is the universal central extension of E(R). Finally, there will be a study of the K-groups K2 and K2,n, the “symbols” in these groups, and an exact sequence relating K1(R) and K2(R). Throughout this chapter, the ring R is understood to be arbitrary unless specific additional assumptions are made.