semidirect

Semidirect refers to a construction in algebra in which two structures are combined using an action of one on the other. In group theory, the semidirect product G ⋊φ H is a way to assemble two groups G and H into a new group using a homomorphism φ: H → Aut(G), the group of automorphisms of G.

As a set, the semidirect product is the Cartesian product G × H, with multiplication defined by (g1, h1)(g2, h2) = (g1 φ(h1)(g2), h1 h2). The subgroup G0 = G × {e} is normal in G ×φ H, and the subgroup H0 = {e} × H is a complement of G0. The projection onto H gives a surjective homomorphism from the semidirect product to H with kernel G, so the short exact sequence 1 → G → G ×φ H → H → 1 splits. If φ is the trivial homomorphism, G ×φ H is simply the direct product G × H.

The action φ encodes how H twists G; a nontrivial action can yield non-abelian groups. Classic examples include the dihedral group D4 of order 8, realized as Z4 ⋊ Z2 with the nontrivial automorphism, and the symmetric group S3, realized as Z3 ⋊ Z2 with the nontrivial automorphism of Z3.

Semidirect products generalize to other contexts, such as monoids, Lie algebras, and modules, where the second structure acts by automorphisms or linear transformations on the first. In group theory, semidirect products describe all split group extensions: every split extension is isomorphic to a semidirect product determined by the induced action.