Elementary subgroup

Summary

The elementary subgroup $E_n(R)$ is a subgroup of the general linear group $GL_n(R)$ over a ring $R$. It is generated by the elementary matrices that add a multiple of one row to another. The union $E(R)$ inside the stable general linear group is, by Whitehead's lemma, the commutator subgroup of $GL(R)$. So K1 of a ring is the quotient $GL(R)/E(R)$, which measures how far invertible matrices are from products of row operations.