Direct product

Produit direct (Fr). Direktes Produkt (Ge). Прямое произведение групп (Ru). Prodotto diretto (It). 直積 (Ja). Producto directo (Sp).

In group theory, the direct product of two groups (G, *) and (H, o), denoted by G × H, is the set of the elements obtained by taking the Cartesian product of the sets of elements of G and H: {(g, h): g ∈ G, h ∈ H};

For abelian groups which are written additively, it may also be called the direct sum of two groups, denoted by [math]G \oplus H[/math].

The group obtained in this way has a normal subgroup isomorphic to G [given by the elements of the form (g, 1)], and one isomorphic to H [comprising the elements (1, h)].

The reverse also holds: if a group K contains two normal subgroups G and H, such that K= GH and the intersection of G and H contains only the identity, then K = G × H. A relaxation of these conditions gives the semidirect product.