be the stable general linear group under the stabilization maps $A\mapsto \operatorname{diag}(A,1)$, and let $E(R)\le GL(R)$ be the stable elementary subgroup generated by the elementary transvections. With the classical definition
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\begin{align*}
K_1(R)=GL(R)/E(R),
\end{align*}
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the quotient map $GL(R)\to K_1(R)$ identifies $K_1(R)$ with the abelianization of $GL(R)$. Equivalently, the induced homomorphism