In group theory one defines the direct product of two groups G and H, denoted by G×H, as follows:
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, C_{2}: say {1, a} and {1, b}. Then C_{2}×C_{2} = {(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,b^{2}) = (1,1).
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