Direct product

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

In group theory, direct product of two groups (G, *) and (H, o), denoted by G × H is the as set of the elements obtained by taking the cartesian product of the sets of elements of G and H: {(g, h): g in G, h in 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 x H. A relaxation of these conditions gives the semidirect product.