# Direct product

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 .

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

The *K* contains two normal *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.