split extension

《Abstract Algebra》 · 第 384 页 · Section 10.5「Exact Sequences—Projective, Injective, and Flat Modules」

If $1\to A\xrightarrow{\psi}B\xrightarrow{\varphi}C\to1$ is a short exact sequence of groups, then the sequence is said to be split if there is a subgroup complement to $\psi(A)$ in $B$. In this case, up to isomorphism, $B=A\rtimes C$ (more precisely, $B=\psi(A)\rtimes C'$ for some subgroup $C'$, and $C'$ is mapped isomorphically onto $C$ by $\varphi:\varphi(C')\cong C$). In either case the extension $B$ is said to be a split extension of $C$ by $A$.