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 $G$.

Proof sketch. Exactness implies $N=\ker(\pi)$, hence $N\lhd G$ by kernel is normal . Define a map

Using the definition of the semidirect product multiplication (twisted by $\varphi$) and the fact that $s$ is a homomorphism, check $\Phi$ is a homomorphism. Surjectivity follows because $\pi(g)\in Q$ and $g\,s(\pi(g))^{-1}\in \ker(\pi)=N$. Injectivity and uniqueness come from $N\cap s(Q)=\{e\}$ (a consequence of $\pi\circ s=\mathrm{id}_Q$) and the factorization argument.