1.33: 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 \(G \oplus H\).
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.