Skip to main content
Chemistry LibreTexts

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: {(gh): g in Gh 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 KGH 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.