Direct Product of Groups - Algebraic Structure

Algebraic Structure

Let G and H be groups, let P = G × H, and consider the following two subsets of P:

G' = { (g, 1) : g ∈ G } and H' = { (1, h) : h ∈ H }

Both of these are in fact subgroups of P, the first being isomorphic to G, and the second being isomorphic to H. If we identify these with G and H, respectively, then we can think of the direct product P as containing the original groups G and H as subgroups.

These subgroups of P have the following three important properties: (Saying again that we identify G' and H' with G and H, respectively.)

1. The intersection G ∩ H is trivial.
2. Every element of P can be expressed as the product of an element of G and an element of H.
3. Every element of G commutes with every element of H.

Together, these three properties completely determine the algebraic structure of the direct product P. That is, if P is any group having subgroups G and H that satisfy the properties above, then P is necessarily isomorphic to the direct product of G and H. In this situation, P is sometimes referred to as the internal direct product of its subgroups G and H.

In some contexts, the third property above is replaced by the following:

3'. Both G and H are normal in P.

This property is equivalent to property 3, since the elements of two normal subgroups with trivial intersection necessarily commute, a fact which can be deduced by considering the commutator of any g in G, h in H.