In mathematics, specifically in group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted . 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.
In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted . Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups.
Definition
Given groups (with operation ) and (with operation ), the direct product is defined as follows:
The resulting algebraic object satisfies the axioms for a group. Specifically: Associativity The binary operation on is associative. Identity The direct product has an identity element, namely , where is the identity element of and is the identity element of . Inverses The inverse of an element of is the pair , where is the inverse of in , and is the inverse of in .
Examples
Let be the group of real numbers under addition. Then the direct product is the group of all two-component vectors under the operation of vector addition: . Let be the group of positive real numbers under multiplication. Then the direct product is the group of all vectors in the first quadrant under the operation of component-wise multiplication . Let and be cyclic groups with two elements each:
Then the direct product is isomorphic to the Klein four-group:
Elementary properties
Algebraic structure
Let and be groups, let , and consider the following two subsets of :
and .
Both of these are in fact subgroups of , the first being isomorphic to , and the second being isomorphic to . If we identify these with and , respectively, then we can think of the direct product as containing the original groups and as subgroups.
These subgroups of have the following three important properties: (Saying again that we identify and with and , respectively.)
The intersection is trivial. Every element of can be expressed uniquely as the product of an element of and an element of . Every element of commutes with every element of . Together, these three properties completely determine the algebraic structure of the direct product . That is, if is any group having subgroups and that satisfy the properties above, then is necessarily isomorphic to the direct product of and . In this situation, is sometimes referred to as the internal direct product of its subgroups and .
In some contexts, the third property above is replaced by the following: 3′. Both and are normal in . 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 in , in .
Examples
Presentations
The algebraic structure of can be used to give a presentation for the direct product in terms of the presentations of and . Specifically, suppose that
and
where and are (disjoint) generating sets and and are defining relations. Then
where is a set of relations specifying that each element of commutes with each element of .
For example if
and
then
Normal structure
As mentioned above, the subgroups and are normal in . Specifically, define functions and by
and .
Then and are homomorphisms, known as projection homomorphisms, whose kernels are and , respectively.
It follows that is an extension of by (or vice versa). In the case where is a finite group, it follows that the composition factors of are precisely the union of the composition factors of and the composition factors of .
Further properties
Universal property
The direct product can be characterized by the following universal property. Let and be the projection homomorphisms. Then for any group and any homomorphisms and , there exists a unique homomorphism making the following diagram commute:
Specifically, the homomorphism is given by the formula
. This is a special case of the universal property for products in category theory.
Subgroups
If is a subgroup of and is a subgroup of , then the direct product is a subgroup of . For example, the isomorphic copy of in is the product , where is the trivial subgroup of .
If and are normal, then is a normal subgroup of . Moreover, the quotient of the direct products is isomorphic to the direct product of the quotients: .