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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, specifically in
group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
, the direct product is an operation that takes two
groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic identi ...
and and constructs a new group, usually denoted . This operation is the group-theoretic analogue of the
Cartesian product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of sets and is one of several important notions of
direct productIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
in mathematics. In the context of
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
s, the direct product is sometimes referred to as the
direct sum The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
, 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 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 ...
, every finite abelian group can be expressed as the direct sum of
cyclic group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

cyclic group
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 (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. ;Identity: The direct product has an
identity element In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
, namely , where is the identity element of and is the identity element of . ;Inverses: The
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when add ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s under
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

addition
. Then the direct product is the group of all two-component
vectors Vector may refer to: Biology *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector *Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...
under the operation of
vector addition Vector may refer to: Biology *Vector (epidemiology) In epidemiology, a disease vector is any living agent that carries and transmits an infectious pathogen to another living organism; agents regarded as vectors are organisms, such as Para ...
: :. *Let be the group of
positive real numbers In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
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 The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

cyclic group
s with two elements each:
Then the direct product is
isomorphic In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
to the
Klein four-group In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity) and in which composing any two of the three ...
:


Elementary properties


Algebraic structure

Let and be groups, let , and consider the following two
subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

subset
s of : :    and    . Both of these are in fact
subgroup In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...
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 The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
is trivial. # Every element of can be expressed uniquely as the product of an element of and an element of . # Every element of
commutes
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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 An audience is a group of people who participate in a show or encounter a work of art A work of art, artwork, art piece, piece of art or art object is an artistic creation ...
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
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 In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...
, 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 category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
. Let and be the projection homomorphisms. Then for any group and any homomorphisms and , there exists a unique homomorphism making the following diagram
commute
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 formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
.


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 Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne' ...
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 Goursat's lemma, named after the France, French mathematician Édouard Goursat, is an algebraic theorem about subgroups of the Direct product of groups, direct product of two Group (mathematics), groups. It can be stated more generally in a Goursa ...
. Other subgroups include fiber products of and .


Conjugacy and centralizers

Two elements and are
conjugate Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the change ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
of is simply the product of the centralizers of and : :  =  . Similarly, the
center Center or centre may refer to: Mathematics *Center (geometry) In geometry, a centre (or center) (from Ancient Greek language, Greek ''κέντρον'') of an object is a point in some sense in the middle of the object. According to the speci ...
of is the product of the centers of and : :  =  .
Normalizer In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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 or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In , a matrix (plural matrices) is a array or table of s, s, or s, arranged in rows and columns, which is used to represent a or a property of such an object. Fo ...
with integer entries and
determinant In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

determinant
, . This automorphism group is infinite, but only finitely many of the automorphisms have the form given above. In general, every
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group ...
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 (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( ...
. 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 :\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 (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
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 The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
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 category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
in the
category of groups In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
.


Subdirect products

If and are groups, a subdirect product of and is any subgroup of which maps
surjectively
surjectively
onto and under the projection homomorphisms. By
Goursat's lemma Goursat's lemma, named after the France, French mathematician Édouard Goursat, is an algebraic theorem about subgroups of the Direct product of groups, direct product of two Group (mathematics), groups. It can be stated more generally in a Goursa ...
, 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 probabl ...
, is the following subgroup of : :  =  . If and are
epimorphism 220px In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labe ...
s, 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 Publishing is the activity of making information, literature, music, software and other content available to the public for sale or for free. ...
, location=Berlin, New York , isbn=978-0-387-94461-6 , year=1996. Group theory