Short exact sequence

It is an exact sequence of the form

It turns out that $C=B/\text{im}(f)$, or, if we think of $f$ like an inclusion,

It is called split if there exist a morphism $h:C\to B$ such that $g\circ h=id$. If these are abelian groups, then being split is equivalent to

Equivalently, the short exact sequence split if there exists $h:B\to A$ with $h\circ f=id$ (see splitting lemma.