Stable general linear group (${\mathrm{GL}}_{\infty }(A)={\underrightarrow{\lim }}_n{\mathrm{GL}}_n(A)$)

The union of finite invertible matrices under $g\mapsto \mathrm{diag}(g,1)$. Every element and relation occurs in a finite matrix size; adding identity blocks permits comparisons between different sizes without changing the stable class.

The subgroup generated by elementary transvections in finite matrix sizes, with identity stabilization. A third distinct index expresses each transvection as a commutator; stable block identities show that this subgroup is normal and that its quotient in the stable general linear group is abelian. Thus it is precisely the commutator subgroup.