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, 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, 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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