Semidirect Product Construction

A group construction that builds a group G as a semidirect product N ⋊φ H from a normal subgroup N, a subgroup H, and a homomorphism φ: H → Aut(N) specifying how H acts on N; elements multiply by (n1,h1)(n2,h2) = (n1 φ(h1)(n2), h1h2).

A construction that combines two algebraic objects (typically a normal object N and a complement H) into a new object N ⋊_φ H using a specified action φ: H → Aut(N); the underlying set is typically the product N × H with multiplication twisted by the action so that H acts by automorphisms on N.