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
A group is a number of persons 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 ide ...
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 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 notation, that is
: A\ti ...
of
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
s and is one of several important notions of
direct product in mathematics.
In the context of
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
s, 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
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
, every finite abelian group can be expressed as the direct sum of
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
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.
;Identity: The direct product has an
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a 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 algebraic structures su ...
, 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 measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
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 (or length) and direction. Vectors can be added to other vectors a ...
:
:.
*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, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
s with two elements each:
Then the direct product is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
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, 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 ''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, ''H'' is a subgroup ...
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
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or forked ...
.
# 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 Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
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. Presenta ...
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
Projection, projections or projective may refer to:
Physics
* Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction
* The display of images by a projector
Optics, graphic ...
homomorphisms, whose kernels are and , respectively.
It follows that is an
extension
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* E ...
of by (or vice versa). In the case where is a
finite group
Finite is the opposite of 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 or marked ...
, it follows that the
composition factor In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natur ...
s of are precisely the
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
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
Commute, commutation or commutative may refer to:
* Commuting, the process of travelling between a place of residence and a place of work
Mathematics
* Commutative property, a property of a mathematical operation whose result is insensitive to th ...
:
:
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, cate ...
.
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
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or forked ...
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 of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of 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 In the mathematical discipline of group theory, for a given group the diagonal subgroup of the ''n''-fold direct product is the subgroup
:\.
This subgroup is isomorphic to
Properties and applications
* If acts on a set the ''n''-fold diag ...
:
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 Gours ...
. 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 ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...
of is simply the product of the centralizers of and :
: = .
Similarly, the
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
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 ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...
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 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), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
with integer entries and
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
, . 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 g ...
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
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 groups based on the semidirect product. It is formed by the action of one group on many copies of another group, somewhat analogous to exponentiation. Wreath products are used i ...
. This is part of the
Krull–Schmidt theorem
In mathematics, the Krull–Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.
Definitions
We say that a group ''G ...
, 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
In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a ''family of real numbers, indexed by the set of integers'' is a collection of real numbers, whe ...
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
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or forked ...
.
# Every element of can be expressed uniquely as the product of an element of and an element of .
# Both and are
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
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
In mathematics, especially group theory, the Zappa–Szép product (also known as the Zappa–Rédei–Szép product, general product, knit product, exact factorization or bicrossed product) describes a way in which a group can be constructed fro ...
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, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coprodu ...
in the
category of groups
In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.
Relation to other categories
There a ...
.
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
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 Gours ...
, 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.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: i ...
, is the following subgroup of :
: =
.
If and are
epimorphism
In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms ,
: g_1 \circ f = g_2 \circ f ...
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 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