A normal factor and a complementary subgroup with trivial intersection form a semidirect product.Suggest a slogan
Let \(G\) be a group with subgroups \(N \normal G\) and \(H\leq G\) such that \(G = NH\) and \(N\intersect H=\theset{e}\). Let \(\psi\colon H \to \Aut(N)\) be the conjugation action, \(\psi(h)=h(\wait)h^{-1}\). Then \(G \cong N \semidirect_\psi H\).
If moreover \(H\normal G\), then \(hnh^{-1}n^{-1}\in N\intersect H=\theset{e}\) for all \(n\in N\) and \(h\in H\), so \(\psi\) is trivial and \(G\cong N\times H\).