Direct Product - Group Direct Product

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)