In mathematics, specifically in group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H. This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.
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: 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 [g,h] of any g in G, h in H. The algebraic structure of G × H can be used to give a presentation for the direct product in terms of the presentations of G and H. Specifically, suppose that where
For example if then As mentioned above, the subgroups G and H are normal in G × H. Specifically, define functions πG: G × H → G and πH: G × H → H by Then πG and πH are homomorphisms, known as projection homomorphisms, whose kernels are H and G, respectively.
Then for any group P and any homomorphisms ƒG: P → G and ƒH: P → H, there exists a unique homomorphism ƒ: P → G × H making the following diagram commute: Specifically, the homomorphism ƒ is given by the formula This is a special case of the universal property for products in category theory.
This is part of the Krull–Schmidt theorem, and holds more generally for finite direct products.
Given a finite sequence G1, ..., Gn of groups, the direct product
is defined as follows:This has many of the same properties as the direct product of two groups, and can be characterized algebraically in a similar way.
It is also possible to take the direct product of an infinite number of groups.
Instead, these subgroups generate a subgroup of the direct product known as the infinite direct sum, which consists of all elements that have only finitely many non-identity components.
Recall that a group P with subgroups G and H is isomorphic to the direct product of G and H as long as it satisfies the following three conditions: A semidirect product of G and H is obtained by relaxing the third condition, so that only one of the two subgroups G, H is required to be normal.
The resulting product still consists of ordered pairs (g, h), but with a slightly more complicated rule for multiplication.
In fact, the free product of any two nontrivial groups is infinite.
If G and H are groups, a subdirect product of G and H is any subgroup of G × H which maps surjectively onto G and H under the projection homomorphisms.