Direct product

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In mathematics, one can often define a direct product of objects already known, giving a new one.

For instance, in group theory (but similar constructions hold for rings and other algebraic structures) one defines the direct product of two groups G and H, 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*h' )
    (here we denote, as usual, with "*" the operations of G and of H, as well as the new one we are defining).

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).