K1 of a ring

Summary

The group $K_1(R)$ of a ring $R$ is the abelianisation of the stable general linear group $GL(R)$. By Whitehead's lemma it is the quotient of $GL(R)$ by the elementary subgroup $E(R)$. It generalises the determinant: for a commutative ring the determinant induces a split surjection $K_1(R) \to R^\times$, and for a field this is an isomorphism. It is the first of the algebraic K-groups above $K_0$ and carries the Whitehead group used in topology.