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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically in
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, 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 In mathematics, specifically set theory, the Cartesian product of two Set (mathematics), sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notatio ...
of sets and is one of several important notions of
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying Set (mathematics), sets, together with a suitably defined structure on the product set. More ...
in mathematics. In the context of
abelian group In mathematics, an abelian group, also called a commutative group, is a group (mathematics), group in which the result of applying the group Operation (mathematics), operation to two group elements does not depend on the order in which they are w ...
s, the direct product is sometimes referred to as the
direct sum The direct sum is an Operation (mathematics), operation between Mathematical structure, structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct ...
, 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 group In group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathemati ...
s.


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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
. ;Identity: The direct product has an
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a Set (mathematics), set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in alge ...
, 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 number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s under
addition Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
. Then the direct product is the group of all two-component vectors under the operation of
vector addition In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has Magnitude (mathematics), magnitude (or euclidean norm, length) and Direction ( ...
: :. *Let be the group of
positive real numbers In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
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 group In group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathemati ...
s with two elements each:
Then the direct product is
isomorphic In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
to the Klein four-group:


Elementary properties


Algebraic structure

Let and be groups, let , and consider the following two
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s of : :    and    . Both of these are in fact
subgroup In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ...
s 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 In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, a ...
of any in , in .


Examples


Presentations

The algebraic structure of can be used to give a
presentation A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presen ...
for the direct product in terms of the presentations of and . Specifically, suppose that :G = \langle S_G \mid R_G \rangle \ \ and \ \ H = \langle S_H \mid R_H \rangle, where S_G and S_H are (disjoint) generating sets and R_G and R_H are defining relations. Then :G \times H = \langle S_G \cup S_H \mid R_G \cup R_H \cup R_P \rangle 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 = \langle a \mid a^3=1 \rangle \ \ and \ \ H = \langle b \mid b^5=1 \rangle then :G \times H = \langle a, b \mid a^3 = 1, b^5 = 1, ab=ba \rangle.


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 Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
, 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 In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
. 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 Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...
.


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 In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division (mathematics), division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to a ...
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 In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group (mathematics), group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutativity, com ...
of is simply the product of the centralizers of and : :  =  . Similarly, the center of is the product of the centers of and : :  =  .
Normalizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group (mathematics), group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutativity, com ...
s 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 In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of map (mathematics), mapping the object to itself while preserving all of its structure. The Set (m ...
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 Matrix most commonly refers to: * The Matrix (franchise), ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within Th ...
with integer entries and
determinant In mathematics, the determinant is a Scalar (mathematics), scalar value that is a function (mathematics), function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In p ...
, . This automorphism group is infinite, but only finitely many of the automorphisms have the form given above. In general, every
endomorphism In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
of can be written as a matrix :\begin\alpha & \beta \\ \gamma & \delta\end 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 An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
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 In group theory, the wreath product is a special combination of two Group (mathematics), groups based on the semidirect product. It is formed by the Action (group theory), action of one group on many copies of another group, somewhat analogous to ...
. This is part of the
Krull–Schmidt theorem In mathematics, the Krull–Schmidt theorem states that a group (mathematics), group subjected to certain finite set, finiteness conditions on chain (order theory), chains of subgroups, can be uniquely written as a finite direct product of groups, ...
, 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 :\prod_^n G_i \;=\; G_1 \times G_2 \times \cdots \times G_n 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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
.


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 In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a Pushforward (disambiguation), pushforward. Precomposition Precomposition with a Function (mathematics), function probab ...
, is the following subgroup of : :  =  . If and are epimorphisms, then this is a subdirect product.


References

* * . * . * * . * {{Citation , last1=Robinson , first1=Derek John Scott , title=A course in the theory of groups , publisher=
Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...
, location=Berlin, New York , isbn=978-0-387-94461-6 , year=1996. Group products