semidirekte

Semidirekte, or semidirect product, is a construction in group theory used to build a new group from two given groups by specifying how one group acts on the other. It enables the combination of a normal structure with a complementary subgroup to form a larger, often non-abelian, group.

External semidirect product. Let G and H be groups and let φ: H → Aut(G) be a homomorphism into the automorphism group of G. The external semidirect product G ⋊φ H is the set G × H with multiplication

(g1, h1) · (g2, h2) = (g1 φ(h1)(g2), h1 h2).

The identity is (e_G, e_H) and the inverse is (g, h)^{-1} = (φ(h^{-1})(g^{-1}), h^{-1}).

If φ is the trivial homomorphism, G ⋊φ H is isomorphic to the direct product G × H.

Internal semidirect product. Suppose a group E contains a normal subgroup N and a subgroup H such that E = NH and N ∩ H = {e}. The action of H on N by conjugation, hnh^{-1}, defines a homomorphism φ: H → Aut(N). Then E is isomorphic to the external semidirect product N ⋊φ H. This construction shows how the larger group E encodes both the normal structure N and the way H acts on it.

Examples. The symmetric group S3 is isomorphic to C3 ⋊φ C2 with a nontrivial φ, reflecting the action of a transposition on 3-cycles. The dihedral group D4 (order 8) is isomorphic to C4 ⋊φ C2 where the nontrivial element of C2 inverts elements of C4.

Notes. If φ is trivial, the semidirect product reduces to the direct product. The concept extends to other algebraic structures and to Lie groups, with corresponding action maps.