Direct product

In mathematics, one can often define a direct product of objects already known, giving a new one. Examples are the product of groups (described below), the product of rings and of other algebraic structures. The product of topological spaces is another instance.

In group theory one defines the direct product of two groups (G, *) and (H, o) , denoted by G×H, as follows:

• as set of the elements of the new group, take the cartesian product of the sets of elements of G and H, that is {(g, h): g in G, h in H};
• on these elements put an operation, defined elementwise: (g, h) × (g' , h' ) = (g * g' , h o h' )

(Note the operation * may be the same as o.)

This construction gives a new group. It 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)).

As an example, take as G and H two copies of the unique (up to isomorphisms) group of order 2, C₂: say {1, a} and {1, b}. Then C₂×C₂ = {(1,1), (1,b), (a,1), (a,b)}, with the operation element by element. For instance, (1,b)*(a,1) = (1*a, b*1) = (a,b), and (1,b)*(1,b) = (1,b²) = (1,1).