Normal complementary factors with trivial intersection form a direct product.Suggest a slogan
Let \(G\) be a group with subgroups \(H\) and \(K\) such that
\(H, K \normal G\),
\(G = HK\), and
\(H\intersect K = \theset{e}\).
Then the map \(H \times K\to G\), \((h,k)\mapsto hk\), is an isomorphism, so \(G \cong H \times K\).
Hypothesis 1 may be replaced by the condition that \(hk=kh\) for all \(h\in H\) and \(k\in K\): together with \(G=HK\), this condition implies that \(H\) and \(K\) are normal in \(G\).