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Direct product

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    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.

    Direct product is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Online Dictionary of Crystallography.

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