= K1 of a ring
{c}
{title2=$K_1(A)=\mathrm{GL}_\infty(A)/E_\infty(A)$}
Stable invertible <matrices>, with stabilization $g\mapsto\operatorname{diag}(g,1)$, form the <stable general linear group>. Its elementary subgroup is its <commutator subgroup>, and the displayed quotient is its <abelianization>. <Elementary matrices> lift along any surjective <ring> map, even when arbitrary invertible <matrices> do not lift. Over a <field>, <determinant> identifies $K_1$ with its multiplicative unit <group>.
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