= Section normal form for a split group extension
{title2=$g=i(\pi(g))n,\quad n\in N$}
If a <group homomorphism> section $i:G/N\to G$ satisfies $\pi i=\mathrm{id}$, then each $g\in G$ has a unique form $i(q)n$ with $q\in G/N$ and $n\in N$: take $q=\pi(g)$ and $n=i(q)^{-1}g$. Thus $(q,n)\mapsto i(q)n$ is a bijection. It is a homomorphism for the ordinary <direct product of groups> exactly when the section image centralizes $N$. In general it describes a <semidirect product> with a nontrivial conjugation action.
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