For a unital ring $A$ and $n\in\mathbb N$, let $M_n(A)$ be the ring of $n\times n$ matrices over $A$ and let $GL_n(A)=M_n(A)^\times$ be its group of units. Let $\sigma_{n,A}:GL_n(A)\to GL_{n+1}(A)$ be the stabilization homomorphism $X\mapsto \operatorname{diag}(X,1_A)$, and let $GL(A)=\varinjlim_n GL_n(A)$ with respect to these maps. Let $E(A)\trianglelefteq GL(A)$ be the stable elementary subgroup generated by the images of all elementary transvections $e_{ij}(a)=I_n+aE_{ij}$, where $n\ge2$, $1\le i\ne j\le n$, $a\in A$, and $E_{ij}\in M_n(A)$ has $1_A$ in the $(i,j)$-entry and $0_A$ elsewhere. Define $K_1(A)=GL(A)/E(A)$. If $R$ and $S$ are unital rings and $\varphi:R\to S$ is a unital ring homomorphism, then entrywise application of $\varphi$ to representatives in finite general linear groups induces a well-defined [group homomorphism](/page/Group%20Homomorphism) $K_1(\varphi):K_1(R)\to K_1(S)$. For every unital ring $R$, $K_1(\operatorname{id}_R)=\operatorname{id}_{K_1(R)}$. For unital ring homomorphisms $\varphi:R\to S$ and $\psi:S\to T$, $K_1(\psi\circ\varphi)=K_1(\psi)\circ K_1(\varphi)$.