Definition. complementary subgroups [fgap-000H]
Definition. complementary subgroups [fgap-000H]
Subgroups \(N,H\leq G\) are complementary, in this ordered sense, when multiplication \[ N\times H\longrightarrow G,\qquad (n,h)\longmapsto nh \] is a bijection. Equivalently, every \(g\in G\) has a unique factorization \(g=nh\).
For subgroups, the two conditions \[ N\cap H=\{1\},\qquad NH=G \] give complementarity. Existence follows from \(NH=G\). For uniqueness, if \(n_1h_1=n_2h_2\), then \[ n_2^{-1}n_1=h_2h_1^{-1}\in N\cap H, \] so both sides are \(1\).
Trivial intersection alone is not enough: a complement must also provide the factorization of the ambient group. Fré uses these ingredients for an internal semidirect product in [fre2023discrete, sec. 4.2.13, pp. 62--63].