Direct product of groups

In mathematics, specifically in group theory, the direct product is an operation that takes two groups Template:Math and Template:Math and constructs a new group, usually denoted Template:Math. 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 $G\oplus H$. 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 Template:Math(with operation Template:Math) and Template:Math (with operation Template:Math), the direct product Template:Math is defined as follows:Template:Ordered list

The resulting algebraic object satisfies the axioms for a group. Specifically:

Associativity

The binary operation on Template:Math is associative.

Identity

The direct product has an identity element, namely Template:Math, where Template:Math is the identity element of Template:Math and Template:Math is the identity element of Template:Math.

Inverses

The inverse of an element Template:Math of Template:Math is the pair Template:Math, where Template:Math is the inverse of Template:Math in Template:Math, and Template:Math is the inverse of Template:Math in Template:Math.

Examples

• Let Template:Math be the group of real numbers under addition. Then the direct product Template:Math is the group of all two-component vectors Template:Math under the operation of vector addition:

Template:Math.

• Let Template:Math be the group of positive real numbers under multiplication. Then the direct product Template:Math is the group of all vectors in the first quadrant under the operation of component-wise multiplication

Template:Math.

• Let Template:Math and Template:Math be cyclic groups with two elements each:

• $G$

  *	1	a
  1	1	a
  a	a	1

• $H$

  *	1	b
  1	1	b
  b	b	1

Then the direct product Template:Math is isomorphic to the Klein four-group:

$G\times H$

*	(1,1)	(a,1)	(1,b)	(a,b)
(1,1)	(1,1)	(a,1)	(1,b)	(a,b)
(a,1)	(a,1)	(1,1)	(a,b)	(1,b)
(1,b)	(1,b)	(a,b)	(1,1)	(a,1)
(a,b)	(a,b)	(1,b)	(a,1)	(1,1)

Algebraic structure

Let Template:Math and Template:Math be groups, let Template:Math, and consider the following two subsets of Template:Math:

Template:Math    and    Template:Math.

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

These subgroups of Template:Math have the following three important properties: (Saying again that we identify Template:Math and Template:Math with Template:Math and Template:Math, respectively.)

1. The intersection Template:Math is trivial.
2. Every element of Template:Math can be expressed uniquely as the product of an element of Template:Math and an element of Template:Math.
3. Every element of Template:Math commutes with every element of Template:Math.

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

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

3′.  Both Template:Math and Template:Math are normal in Template:Math.

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 Template:Math of any Template:Math in Template:Math, Template:Math in Template:Math.

Presentations

The algebraic structure of Template:Math can be used to give a presentation for the direct product in terms of the presentations of Template:Math and Template:Math. Specifically, suppose that

$G=⟨S_G\mid R_G⟩$ and $H=⟨S_H\mid R_H⟩,$

where $S_G$ and $S_H$ are (disjoint) generating sets and $R_G$ and $R_H$ are defining relations. Then

$G\times H=⟨S_G\cup S_H\mid R_G\cup R_H\cup R_P⟩$

where $R_P$ is a set of relations specifying that each element of $S_G$ commutes with each element of $S_H$.

For example if

$G=⟨a\mid a^3=1⟩$ and $H=⟨b\mid b^5=1⟩$

then

$G\times H=⟨a,b\mid a^3=1,b^5=1,ab=ba⟩.$

Normal structure

As mentioned above, the subgroups Template:Math and Template:Math are normal in Template:Math. Specifically, define functions Template:Math and Template:Math by

Template:Math     and     Template:Math.

Then Template:Math and Template:Math are homomorphisms, known as projection homomorphisms, whose kernels are Template:Math and Template:Math, respectively.

It follows that Template:Math is an extension of Template:Math by Template:Math (or vice versa). In the case where Template:Math is a finite group, it follows that the composition factors of Template:Math are precisely the union of the composition factors of Template:Math and the composition factors of Template:Math.

Further properties

Universal property

The direct product Template:Math can be characterized by the following universal property. Let Template:Math and Template:Math be the projection homomorphisms. Then for any group Template:Math and any homomorphisms Template:Math and Template:Math, there exists a unique homomorphism Template:Math making the following diagram commute:

File:DirectProductDiagram.png

Specifically, the homomorphism Template:Math is given by the formula

Template:Math.

This is a special case of the universal property for products in category theory.

Subgroups

If Template:Math is a subgroup of Template:Math and Template:Math is a subgroup of Template:Math, then the direct product Template:Math is a subgroup of Template:Math. For example, the isomorphic copy of Template:Math in Template:Math is the product Template:Math, where Template:Math is the trivial subgroup of Template:Math.

If Template:Math and Template:Math are normal, then Template:Math is a normal subgroup of Template:Math. Moreover, the quotient of the direct products is isomorphic to the direct product of the quotients:

Template:Math.

Note that it is not true in general that every subgroup of Template:Math is the product of a subgroup of Template:Math with a subgroup of Template:Math. For example, if Template:Math is any non-trivial group, then the product Template:Math has a diagonal subgroup

Template:Math

which is not the direct product of two subgroups of Template:Math.

The subgroups of direct products are described by Goursat's lemma. Other subgroups include fiber products of Template:Math and Template:Math.

Conjugacy and centralizers

Two elements Template:Math and Template:Math are conjugate in Template:Math if and only if Template:Math and Template:Math are conjugate in Template:Math and Template:Math and Template:Math are conjugate in Template:Math. It follows that each conjugacy class in Template:Math is simply the Cartesian product of a conjugacy class in Template:Math and a conjugacy class in Template:Math.

Along the same lines, if Template:Math, the centralizer of Template:Math is simply the product of the centralizers of Template:Math and Template:Math:

Template:Math  =  Template:Math.

Similarly, the center of Template:Math is the product of the centers of Template:Math and Template:Math:

Template:Math  =  Template:Math.

Normalizers behave in a more complex manner since not all subgroups of direct products themselves decompose as direct products.

Automorphisms and endomorphisms

If Template:Math is an automorphism of Template:Math and Template:Math is an automorphism of Template:Math, then the product function Template:Math defined by

Template:Math

is an automorphism of Template:Math. It follows that Template:Math has a subgroup isomorphic to the direct product Template:Math.

It is not true in general that every automorphism of Template:Math has the above form. (That is, Template:Math is often a proper subgroup of Template:Math.) For example, if Template:Math is any group, then there exists an automorphism Template:Math of Template:Math that switches the two factors, i.e.

Template:Math.

For another example, the automorphism group of Template:Math is Template:Math, the group of all Template:Math matrices with integer entries and determinant, Template:Math. This automorphism group is infinite, but only finitely many of the automorphisms have the form given above.

In general, every endomorphism of Template:Math can be written as a Template:Math matrix

$\left[\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right]$

where Template:Math is an endomorphism of Template:Math, Template:Math is an endomorphism of Template:Math, and Template:Math and Template:Math are homomorphisms. Such a matrix must have the property that every element in the image of Template:Math commutes with every element in the image of Template:Math, and every element in the image of Template:Math commutes with every element in the image of Template:Math.

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 Template:Math of groups, the direct product

$\prod _{i=1}^n{G}_i=G_1\times G_2\times \cdots \times G_n$

is defined as follows:Template:Unordered list

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 Template:Math 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 { Template:Math }Template:Math of groups, the direct product Template:Math is defined as follows:Template:Unordered list

Unlike a finite direct product, the infinite direct product Template:Math is not generated by the elements of the isomorphic subgroups { Template:Math }Template:Math. 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 Template:Math with subgroups Template:Math and Template:Math is isomorphic to the direct product of Template:Math and Template:Math as long as it satisfies the following three conditions:

1. The intersection Template:Math is trivial.
2. Every element of Template:Math can be expressed uniquely as the product of an element of Template:Math and an element of Template:Math.
3. Both Template:Math and Template:Math are normal in Template:Math.

A semidirect product of Template:Math and Template:Math is obtained by relaxing the third condition, so that only one of the two subgroups Template:Math is required to be normal. The resulting product still consists of ordered pairs Template:Math, 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 Template:Math is referred to as a Zappa–Szép product of Template:Math and Template:Math.

Free products

The free product of Template:Math and Template:Math, usually denoted Template:Math, is similar to the direct product, except that the subgroups Template:Math and Template:Math of Template:Math are not required to commute. That is, if

Template:Math = Template:Math|Template:Math     and     Template:Math = Template:Math|Template:Math,

are presentations for Template:Math and Template:Math, then

Template:Math = Template:Math|Template:Math.

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 Template:Math and Template:Math are groups, a subdirect product of Template:Math and Template:Math is any subgroup of Template:Math which maps surjectively onto Template:Math and Template:Math under the projection homomorphisms. By Goursat's lemma, every subdirect product is a fiber product.

Fiber products

Let Template:Math, Template:Math, and Template:Math be groups, and let Template:Math and Template:Math be homomorphisms. The fiber product of Template:Math and Template:Math over Template:Math, also known as a pullback, is the following subgroup of Template:Math:

$$G{\times }_{Q}H=\{(g,h)\in G\times H:\phi (g)=\chi (h)\}\text{.}$$ If Template:Math and Template:Math are epimorphisms, then this is a subdirect product.

References

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• Template:Lang Algebra
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