It is an exact sequence of the form
$$ 0 \longrightarrow A \stackrel{f}{\longrightarrow}B \stackrel{g}{\longrightarrow} C \longrightarrow 0. $$It turns out that $C=B/\mbox{im}(f)$, or, if we think of $f$ like an inclusion,
$$ C\cong B/A. $$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
$$ B\cong A\oplus C. $$Equivalently, the short exact sequence split if there exists $h:B\to A$ with $h\circ f=id$ (see splitting lemma.
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Author of the notes: Antonio J. Pan-Collantes
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