Group Direct Product
In group theory one can define the direct product of two groups (G, *) and (H, ●), denoted by G × H. For abelian groups which are written additively, it may also be called the direct sum of two groups, denoted by .
It is defined as follows:
- the set of the elements of the new group is 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' )
(Note the operation * may be the same as ●.)
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)).
The reverse also holds, there is the following recognition theorem: 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 is isomorphic to G x H. A relaxation of these conditions, requiring only one subgroup to be normal, gives the semidirect product.
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).
With a direct product, we get some natural group homomorphisms for free: the projection maps
- ,
called the coordinate functions.
Also, every homomorphism f on the direct product is totally determined by its component functions .
For any group (G, *), and any integer n ≥ 0, multiple application of the direct product gives the group of all n-tuples Gn (for n=0 the trivial group). Examples:
- Zn
- Rn (with additional vector space structure this is called Euclidean space, see below)
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