Let \[ 1 \to A \mapsvia{f} B \mapsvia{g} C \to 1 \] be a short exact sequence of groups. The sequence splits if there exists a homomorphism \(s\colon C\to B\) with \(g\circ s = \id_C\).
For a short exact sequence \(0\to A\mapsvia{f} B\mapsvia{g} C\to 0\) of abelian groups, or more generally of modules over a ring, the following are equivalent: the sequence splits; there exists a homomorphism \(r\colon B\to A\) with \(r\circ f = \id_A\); there exists an isomorphism \(B\cong A\oplus C\) under which \(f\) is the inclusion of \(A\) and \(g\) is the projection onto \(C\). For groups, if \(s\) splits the sequence, then \(B \cong A\rtimes_\varphi C\), where \(\varphi\colon C\to\Aut(A)\) is given by \(f(\varphi(c)(a)) = s(c)\,f(a)\,s(c)^{-1}\).
[@DF04].