Direct product of groups

Direct product of groups
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:
.
Note that it is not true in general that every subgroup of is the product of a subgroup of with a subgroup of . For example, if is any non-trivial group, then the product has a diagonal subgroup
which is not the direct product of two subgroups of .
The subgroups of direct products are described by Goursat's lemma. Other subgroups include fiber products of and .
Conjugacy and centralizers
Two elements and are conjugate in if and only if and are conjugate in and and are conjugate in . It follows that each conjugacy class in is simply the Cartesian product of a conjugacy class in and a conjugacy class in .
Along the same lines, if , the centralizer of is simply the product of the centralizers of and :
= .
Similarly, the center of is the product of the centers of and :
= .
Normalizers behave in a more complex manner since not all subgroups of direct products themselves decompose as direct products.
Automorphisms and endomorphisms
If is an automorphism of and is an automorphism of , then the product function defined by
is an automorphism of . It follows that has a subgroup isomorphic
to the direct product .
It is not true in general that every automorphism of has the above form. (That is, is often a proper subgroup of .) For example, if is any group, then there exists an automorphism of that switches the two factors, i.e.
.
For another example, the automorphism group of is , the group of all matrices with integer entries and determinant, . This automorphism group is infinite, but only finitely many of the automorphisms have the form given above.
In general, every endomorphism of can be written as a matrix
where is an endomorphism of , is an endomorphism of , and and are homomorphisms. Such a matrix must have the property that every element in the image of commutes with every element in the image of , and every element in the image of commutes with every element in the image of .
When G and H are indecomposable, centerless groups, then the automorphism group is relatively straightforward, being Aut(G) × Aut(H) if G and H are not isomorphic, and Aut(G) wr 2 if G ≅ H, wr denotes the wreath product. This is part of the Krull–Schmidt theorem, and holds more generally for finite direct products.
Generalizations
Finite direct products
It is possible to take the direct product of more than two groups at once. Given a finite sequence 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.
Infinite direct products
It is also possible to take the direct product of an infinite number of groups. For an infinite sequence of groups, this can be defined just like the finite direct product of above, with elements of the infinite direct product being infinite tuples.
More generally, given an indexed family { } of groups, the direct product is defined as follows:
Unlike a finite direct product, the infinite direct product is not generated by the elements of the isomorphic subgroups { }. 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.
Other products
Semidirect products
Recall that a group with subgroups and is isomorphic to the direct product of and as long as it satisfies the following three conditions:
The intersection is trivial.
Every element of can be expressed uniquely as the product of an element of and an element of .
Both and are normal in .
A semidirect product of and is obtained by relaxing the third condition, so that only one of the two subgroups is required to be normal. The resulting product still consists of ordered pairs , but with a slightly more complicated rule for multiplication.
It is also possible to relax the third condition entirely, requiring neither of the two subgroups to be normal. In this case, the group is referred to as a Zappa–Szép product of and .
Free products
The free product of and , usually denoted , is similar to the direct product, except that the subgroups and of are not required to commute. That is, if
= | and = |,
are presentations for and , then
= |.
Unlike the direct product, elements of the free product cannot be represented by ordered pairs. In fact, the free product of any two nontrivial groups is infinite. The free product is actually the coproduct in the category of groups.
Subdirect products
If and are groups, a subdirect product of and is any subgroup of which maps surjectively onto and under the projection homomorphisms. By Goursat's lemma, every subdirect product is a fiber product.
Fiber products
Let and be groups, and let and be homomorphisms. The fiber product of and over , also known as a pullback, is the following subgroup of :
If and are epimorphisms, then this is a subdirect product.
References
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Group products