Semidirect product from a splitting exact sequence

Proposition (Semidirect product from splitting). Let

be an exact sequence of groups . Suppose the sequence splits, meaning there exists a homomorphism $s:Q\to G$ (a section) such that $\pi\circ s=\mathrm{id}_Q$; equivalently, $G$ is a split extension of $Q$ by $N$.

Then:

1. $\iota(N)$ is a normal subgroup of $G$ and we may identify $N$ with $\iota(N)\lhd G$.

2. Let $\varphi:Q\to \mathrm{Aut}(N)$ be the homomorphism defined by

   $$\varphi(q)(n) := s(q)\,n\,s(q)^{-1}\qquad (q\in Q,\ n\in N).$$

   Then $G$ is isomorphic to the semidirect product $N\rtimes_{\varphi} Q$.

3. Under this isomorphism, every $g\in G$ can be written uniquely as $g = n\,s(q)$ with $n\in N$ and $q\in Q$.

Context. This proposition is the standard bridge between abstract extensions and concrete constructions: split exact sequences are exactly semidirect products. The “internal” version is phrased via the internal semidirect product inside $G$.