Direct product

From Online Dictionary of Crystallography

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): gG, hH};

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.