# Direct product

### From Online Dictionary of Crystallography

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

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 .

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.