Semidirektes

Semidirektes (semidirektes Produkt) is a construction in group theory that combines two groups into a new group using a nontrivial action of one group on the other. It generalizes the direct product by allowing the second factor to act by automorphisms on the first.

In the external version, let N and H be groups and let φ: H → Aut(N) be a homomorphism. The external semidirect product N ⋊φ H consists of the set N × H with the multiplication (n1, h1) · (n2, h2) = (n1 φ(h1)(n2), h1 h2). If φ is the trivial homomorphism, this product is simply the direct product N × H. Different choices of φ can yield nonisomorphic semidirect products.

The internal semidirect product describes when a single group G already contains the relevant subgroups. If G has a normal subgroup N ⊲ G and a subgroup H ≤ G such that G = NH and N ∩ H = {e}, then G is the internal semidirect product of N by H. In this case G is isomorphic to the external product N ⋊φ H, where φ is given by conjugation of H on N.

Examples illustrate the concept. The symmetric group S3 is the semidirect product C3 ⋊ C2 with the nontrivial action of C2 on C3 (inverting a 3-cycle). The dihedral group D6 (order 12) is the semidirect product C6 ⋊ C2 with inversion action. A classic infinite example is the Klein bottle group Z ⋊ Z with the nontrivial action of one Z on the other.

Semidirect products are central in studying group extensions, symmetry, and geometry, offering a flexible framework to build and classify groups from simpler components.