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In [[Mathematics|mathematics]], specifically in [[Group theory|group theory]], the '''direct product''' is an operation that takes two [[Group (mathematics)|groups]] {{math|''G''}} and {{math|''H''}} and constructs a new group, usually denoted {{math|''G'' × ''H''}}.  This operation is the group-theoretic analogue of the [[Cartesian product]] of [[Set (mathematics)|set]]s and is one of several important notions of [[Direct product|direct product]] in mathematics.

In the context of [[Abelian group|abelian group]]s, the direct product is sometimes referred to as the [[Direct sum|direct sum]], and is denoted <math>G \oplus H</math>.  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|cyclic group]]s.

== Definition ==

Given groups {{math|''G ''}}(with operation {{math|*}}) and {{math|''H''}} (with operation {{math|∆}}), the '''direct product''' {{math|''G'' × ''H''}} is defined as follows:{{ordered list
| The underlying set is the Cartesian product, {{math|''G'' × ''H''}}. That is, the [[Ordered pair|ordered pair]]s {{math|(''g'', ''h'')}}, where {{math|''g'' ∈ ''G''}} and {{math|''h'' ∈ ''H''}}.
| The [[Binary operation|binary operation]] on {{math|''G'' × ''H''}} is defined component-wise:
:{{math|(''g''<sub>1</sub>, ''h''<sub>1</sub>) · (''g''<sub>2</sub>, ''h''<sub>2</sub>) {{=}} (''g''<sub>1</sub> * ''g''<sub>2</sub>, ''h''<sub>1</sub> ∆ ''h''<sub>2</sub>)}}
}}

The resulting algebraic object satisfies the axioms for a group.  Specifically:
;Associativity: The binary operation on {{math|''G'' × ''H''}} is associative.
;Identity: The direct product has an [[Identity element|identity element]], namely {{math|(1<sub>''G''</sub>, 1<sub>''H''</sub>)}}, where {{math|1<sub>''G''</sub>}} is the identity element of {{math|''G''}} and {{math|1<sub>''H''</sub>}} is the identity element of&nbsp;{{math|''H''}}.
;Inverses: The [[Inverse element|inverse]] of an element {{math|(''g'', ''h'')}} of {{math|''G'' × ''H''}} is the pair {{math|(''g''<sup>−1</sup>, ''h''<sup>−1</sup>)}}, where {{math|''g''<sup>−1</sup>}} is the inverse of {{math|''g''}} in {{math|''G''}}, and {{math|''h''<sup>−1</sup>}} is the inverse of {{math|''h''}} in&nbsp;{{math|''H''}}.

==Examples==

*Let {{math|'''R'''}} be the group of [[Real number|real number]]s under [[Addition|addition]].  Then the direct product {{math|'''R''' × '''R'''}} is the group of all two-component [[Euclidean vector|vectors]] {{math|(''x'', ''y'')}} under the operation of [[Euclidean vector#Addition and subtraction|vector addition]]:
:{{math|(''x''<sub>1</sub>, ''y''<sub>1</sub>) + (''x''<sub>2</sub>, ''y''<sub>2</sub>) &nbsp;{{=}}&nbsp; (''x''<sub>1</sub> + ''x''<sub>2</sub>, ''y''<sub>1</sub> + ''y''<sub>2</sub>)}}.
*Let {{math|'''R'''<sup>+</sup>}} be the group of [[Positive real numbers|positive real numbers]] under multiplication. Then the direct product {{math|'''R'''<sup>+</sup> × '''R'''<sup>+</sup>}} is the group of all  vectors in the first quadrant under the operation of component-wise multiplication
:{{math|(''x''<sub>1</sub>, ''y''<sub>1</sub>) × (''x''<sub>2</sub>, ''y''<sub>2</sub>) &nbsp;{{=}}&nbsp; (''x''<sub>1</sub> × ''x''<sub>2</sub>,&nbsp; ''y''<sub>1</sub> × ''y''<sub>2</sub>)}}.
*Let {{math|''G''}} and {{math|''H''}} be [[Cyclic group|cyclic group]]s with two elements each:
<div><ul>
<li style="display: inline-table;">
{| class="wikitable"
|+ <math>G</math>
|-
! * 
! 1 
! a
|-
! 1 
| 1 || a
|-
! a
| a || 1
|}
</li>
<li style="display: inline-table;margin-left:20px;">
{| class="wikitable"
|+ <math>H</math>
|-
! * 
! 1 
! b
|-
! 1 
| 1 || b
|-
! b
| b || 1
|}
</li>
</ul></div>
Then the direct product {{math|''G'' × ''H''}} is [[Group isomorphism|isomorphic]] to the [[Klein four-group]]:
{| class="wikitable"
|+ <math>G\times H</math>
|-
! * 
! (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)
|}

==Elementary properties==
{{unordered list
| The direct product is commutative and associative up to isomorphism. That is, {{math|''G'' × ''H'' {{large|≅}} ''H'' × ''G''}} and {{math|(''G'' × ''H'') × ''K'' {{large|≅}} ''G'' × (''H'' × ''K'')}} for any groups {{math|''G''}}, {{math|''H''}}, and {{math|''K''}}.

| The [[Trivial group|trivial group]] is the [[Identity element|identity element]] of the direct product, up to isomorphism. If {{math|''E''}} denotes the trivial group, {{math|''G'' {{large|≅}} ''G'' × ''E'' {{large|≅}} ''E'' × ''G''}} for any groups {{math|''G''}}.

| The [[Order (group theory)|order]] of a direct product {{math|''G'' × ''H''}} is the product of the orders of {{math|''G''}} and&nbsp;{{math|''H''}}:
:{{math|{{large|{{!}}}}''G'' × ''H''{{large|{{!}}}} {{=}} {{large|{{!}}}}''G''{{large|{{!}}}}{{thin space}}{{large|{{!}}}}''H''{{large|{{!}}}}}}.

This follows from the formula for the [[Cardinality|cardinality]] of the cartesian product of sets.
| The order of each element {{math|(''g'', ''h'')}} is the [[Least common multiple|least common multiple]] of the orders of {{math|''g''}} and {{math|''h''}}:<ref>{{Cite book|title = Contemporary Abstract Algebra|last = Gallian|first = Joseph A.|publisher = Cengage Learning|year = 2010|isbn = 9780547165097|edition = 7|page = 157}}</ref>
:{{math|{{large|{{!}}}}(''g'', ''h''){{large|{{!}}}} {{=}} lcm{{large|({{!}}}}''g''{{large|{{!}}}}, {{large|{{!}}}}''h''{{large|{{!}})}}}}.

In particular, if {{math|{{large|{{!}}}} ''g'' {{large|{{!}}}}}} and {{math|{{large|{{!}}}} ''h'' {{large|{{!}}}}}} are relatively prime, then the order of {{math| (''g'', ''h'') }} is the product of the orders of {{math| ''g'' }} and {{math| ''h'' }}.
| As a consequence, if {{math|''G''}} and {{math|''H''}} are cyclic groups whose orders are relatively prime, then {{math|''G'' × ''H''}} is cyclic as well.  That is, if {{math|''m''}} and {{math|''n''}} are relatively prime, then
:{{math|('''Z''' / ''m'''''Z''') × ('''Z''' / ''n'''''Z''') {{large|≅}} '''Z''' / ''mn'''''Z'''}}.

This fact is closely related to the [[Chinese remainder theorem]].
}}

==Algebraic structure==
Let {{math|''G''}} and {{math|''H''}} be groups, let {{math|1=''P'' = ''G'' × ''H''}}, and consider the following two [[Subset|subset]]s of&nbsp;{{math|''P''}}:

:{{math|1=''G''′ = { (''g'', 1) : ''g'' ∈ ''G'' } }} &nbsp;&nbsp;&nbsp;and&nbsp;&nbsp;&nbsp; {{math|1=''H''′ = { (1, ''h'') : ''h'' ∈ ''H'' } }}.

Both of these are in fact [[Subgroup|subgroup]]s of {{math|''P''}}, the first being isomorphic to {{math|''G''}}, and the second being isomorphic to {{math|''H''}}.  If we identify these with {{math|''G''}} and {{math|''H''}}, respectively, then we can think of the direct product {{math|''P''}} as containing the original groups {{math|''G''}} and {{math|''H''}} as subgroups.

These subgroups of {{math|''P''}} have the following three important properties:
(Saying again that we identify {{math|''G''′}} and {{math|''H''′}} with {{math|''G''}} and {{math|''H''}}, respectively.)

# The [[Intersection (set theory)|intersection]] {{math|''G'' ∩ ''H''}} is [[Trivial group|trivial]].
# Every element of {{math|''P''}} can be expressed uniquely as the product of an element of {{math|''G''}} and an element of&nbsp;{{math|''H''}}.
# Every element of {{math|''G''}} commutes with every element of {{math|''H''}}.
Together, these three properties completely determine the algebraic structure of the direct product {{math|''P''}}.  That is, if {{math|''P''}} is ''any'' group having subgroups {{math|''G''}} and {{math|''H''}} that satisfy the properties above, then {{math|''P''}} is necessarily isomorphic to the direct product of {{math|''G''}} and {{math|''H''}}.  In this situation, {{math|''P''}} is sometimes referred to as the '''internal direct product''' of its subgroups {{math|''G''}} and {{math|''H''}}.

In some contexts, the third property above is replaced by the following:
:3′. &nbsp;Both {{math|''G''}} and {{math|''H''}} are [[Normal subgroup|normal]] in {{math|''P''}}.
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|commutator]] {{math|[''g'',''h'']}} of any {{math|''g''}} in {{math|''G''}}, {{math|''h''}} in {{math|''H''}}.

===Examples===
{{unordered list
| Let {{math|''V''}} be the [[Klein four-group]]: 
{{aligned table
 | title = {{math|''V''}}
 | class = wikitable
 | style = text-align:center; width:150px; height:150px;
 | cols=5
 | row1header = y | col1header = y
 | ∙              | {{math|1}}     | {{math|''a''}} | {{math|''b''}} | {{math|''c''}}
 | {{math|1}}     | {{math|1}}     | {{math|''a''}} | {{math|''b''}} | {{math|''c''}}
 | {{math|''a''}} | {{math|''a''}} | {{math|1}}     | {{math|''c''}} | {{math|''b''}}
 | {{math|''b''}} | {{math|''b''}} | {{math|''c''}} | {{math|1}}     | {{math|''a''}}
 | {{math|''c''}} | {{math|''c''}} | {{math|''b''}} | {{math|''a''}} | {{math|1}}
}} 
Then {{math|''V''}} is the internal direct product of the two-element subgroups {1, ''a''} and {1, ''b''}.
| Let <math>\langle a \rangle</math> be a cyclic group of order ''mn'', where ''m'' and ''n'' are relatively prime.  Then <math>\langle a^n \rangle</math> and <math>\langle a^m \rangle</math> are cyclic subgroups of orders ''m'' and ''n'', respectively, and <math>\langle a \rangle</math> is the internal direct product of these subgroups.
| Let {{math|'''C'''<sup>×</sup>}} be the group of nonzero [[Complex number|complex number]]s under [[Multiplication|multiplication]].  Then {{math|'''C'''<sup>×</sup>}} is the internal direct product of the [[Circle group|circle group]] {{math|'''T'''}} of unit complex numbers and the group {{math|'''R'''<sup>+</sup>}} of [[Positive real numbers|positive real numbers]] under multiplication.
| If ''n'' is odd, then the [[General linear group|general linear group]] {{math|GL(''n'', '''R''')}} is the internal direct product of the [[Special linear group|special linear group]] {{math|SL(''n'', '''R''')}} and the subgroup consisting of all scalar matrices.
| Similarly, when ''n'' is odd the [[Orthogonal group|orthogonal group]] {{math|O(''n'', '''R''')}} is the internal direct product of the special orthogonal group {{math|SO(''n'', '''R''')}} and the two-element subgroup {{math|{−''I'', ''I''},}} where {{math|''I''}} denotes the [[Identity matrix|identity matrix]].
| The [[Symmetry group|symmetry group]] of a [[Cube|cube]] is the internal direct product of the subgroup of rotations and the two-element group {{math|{−''I'', ''I''},}} where {{math|''I''}} is the identity element and {{math|−''I''}} is the [[Point reflection|point reflection]] through the center of the cube.  A similar fact holds true for the symmetry group of an [[Icosahedron|icosahedron]].
| Let ''n'' be odd, and let D<sub>4''n''</sub> be the [[Dihedral group|dihedral group]] of order 4''n'':
:<math>D_{4n} = \langle r, s \mid r^{2n} = s^2 = 1, sr = r^{-1}s \rangle.</math>
Then D<sub>4''n''</sub> is the internal direct product of the subgroup <math>\langle r^2, s \rangle</math> (which is isomorphic to D<sub>2''n''</sub>) and the two-element subgroup {1, ''r''<sup>''n''</sup>}.
}}

===Presentations===
The algebraic structure of {{math|''G'' × ''H''}} can be used to give a presentation for the direct product in terms of the presentations of {{math|''G''}} and {{math|''H''}}.  Specifically, suppose that

:<math>G = \langle S_G \mid R_G \rangle \ \ </math> and <math>\ \ H = \langle S_H \mid R_H \rangle,</math>

where <math>S_G</math> and <math>S_H</math> are (disjoint) [[Generating set of a group|generating sets]] and <math>R_G</math> and <math>R_H</math> are defining relations. Then

:<math>G \times H = \langle S_G \cup S_H \mid R_G \cup R_H \cup R_P \rangle</math>

where <math>R_P</math> is a set of relations specifying that each element of <math>S_G</math> commutes with each element of <math>S_H</math>.

For example if

:<math>G = \langle a \mid a^3=1 \rangle \ \ </math> and <math>\ \ H = \langle b \mid b^5=1 \rangle</math>

then

:<math>G \times H = \langle a, b \mid a^3 = 1, b^5 = 1, ab=ba \rangle.</math>

===Normal structure===
As mentioned above, the subgroups {{math|''G''}} and {{math|''H''}} are normal in {{math|''G'' × ''H''}}.  Specifically, define functions {{math|''π<sub>G</sub>'': ''G'' × ''H'' → ''G''}} and {{math|''π<sub>H</sub>'': ''G'' × ''H'' → ''H''}} by

:{{math|1=''π<sub>G</sub>''(''g'', ''h'') = ''g''}} &nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp; {{math|1=''π<sub>H</sub>''(''g'', ''h'') = ''h''}}.

Then {{math|''π<sub>G</sub>''}} and {{math|''π<sub>H</sub>''}} are [[Group homomorphism|homomorphisms]], known as '''[[Projection (mathematics)|projection]] homomorphisms''', whose kernels are {{math|''H''}} and {{math|''G''}}, respectively.

It follows that {{math|''G'' × ''H''}} is an [[Group extension|extension]] of {{math|''G''}} by {{math|''H''}} (or vice versa).  In the case where {{math|''G'' × ''H''}} is a [[Finite group|finite group]], it follows that the composition factors of {{math|''G'' × ''H''}} are precisely the [[Union (set theory)|union]] of the composition factors of {{math|''G''}} and the composition factors of {{math|''H''}}.

==Further properties==

===Universal property===
{{main | Product (category theory)}}
The direct product {{math|''G'' × ''H''}} can be characterized by the following [[Universal property|universal property]].  Let {{math|''π<sub>G</sub>'': ''G'' × ''H'' → ''G''}} and {{math|''π<sub>H</sub>'': ''G'' × ''H'' → ''H''}} be the projection homomorphisms.  Then for any group {{math|''P''}} and any homomorphisms {{math|ƒ<sub>''G''</sub>: ''P'' → ''G''}} and {{math|ƒ<sub>''H''</sub>: ''P'' → ''H''}}, there exists a unique homomorphism {{math|ƒ: ''P'' → ''G'' × ''H''}} making the following diagram [[Commutative diagram|commute]]:
:[[Image:DirectProductDiagram.png|200px]]
Specifically, the homomorphism {{math|ƒ}} is given by the formula
:{{math|ƒ(''p'') &nbsp;{{=}}&nbsp; <big>(</big> ƒ<sub>''G''</sub>(''p''),  ƒ<sub>''H''</sub>(''p'') <big>)</big>}}.
This is a special case of the universal property for products in [[Category theory|category theory]].

===Subgroups===
If {{math|''A''}} is a subgroup of {{math|''G''}} and {{math|''B''}} is a subgroup of {{math|''H''}}, then the direct product {{math|''A'' × ''B''}} is a subgroup of {{math|''G'' × ''H''}}.  For example, the isomorphic copy of {{math|''G''}} in {{math|''G'' × ''H''}} is the product {{math|''G'' × {{big|{1} }}}}, where {{math|{{big|{1} }}}} is the [[Trivial group|trivial]] subgroup of {{math|''H''}}.

If {{math|''A''}} and {{math|''B''}} are normal, then {{math|''A'' × ''B''}} is a normal subgroup of {{math|''G'' × ''H''}}.  Moreover, the [[Quotient group|quotient]] of the direct products is isomorphic to the direct product of the quotients:
:{{math|(''G'' × ''H'') / (''A'' × ''B'') {{big|≅}} (''G'' / ''A'') × (''H'' / ''B'')}}.

Note that it is not true in general that every subgroup of {{math|''G'' × ''H''}} is the product of a subgroup of {{math|''G''}} with a subgroup of {{math|''H''}}.  For example, if {{math|''G''}} is any non-trivial group, then the product {{math|''G'' × ''G''}} has a [[Diagonal subgroup|diagonal subgroup]]
:{{math|Δ {{=}} { (''g'', ''g'') : ''g'' ∈ ''G'' } }}

which is not the direct product of two subgroups of {{math|''G''}}.

The subgroups of direct products are described by [[Goursat's lemma]].  Other subgroups include fiber products of {{math|''G''}} and {{math|''H''}}.

===Conjugacy and centralizers===
Two elements {{math|(''g''<sub>1</sub>, ''h''<sub>1</sub>)}} and {{math|(''g''<sub>2</sub>, ''h''<sub>2</sub>)}} are [[Conjugacy class|conjugate]] in {{math|''G'' × ''H''}} if and only if {{math|''g''<sub>1</sub>}} and {{math|''g''<sub>2</sub>}} are conjugate in {{math|''G''}} and {{math|''h''<sub>1</sub>}} and {{math|''h''<sub>2</sub>}} are conjugate in {{math|''H''}}.  It follows that each conjugacy class in {{math|''G'' × ''H''}} is simply the Cartesian product of a conjugacy class in {{math|''G''}} and a conjugacy class in {{math|''H''}}.

Along the same lines, if {{math|(''g'', ''h'') ∈ ''G'' × ''H''}}, the centralizer of {{math|(''g'', ''h'')}} is simply the product of the centralizers of {{math|''g''}} and {{math|''h''}}:

:{{math|''C''<sub>''G''×''H''</sub>(''g'', ''h'')}} &nbsp;=&nbsp; {{math|''C''<sub>''G''</sub>(''g'') × ''C''<sub>''H''</sub>(''h'')}}.

Similarly, the [[Center (group theory)|center]] of {{math|''G'' × ''H''}} is the product of the centers of {{math|''G''}} and {{math|''H''}}:

:{{math|''Z''(''G'' × ''H'')}} &nbsp;=&nbsp; {{math|''Z''(''G'') × ''Z''(''H'')}}.

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

===Automorphisms and endomorphisms===
If {{math|''α''}} is an automorphism of {{math|''G''}} and {{math|''β''}} is an automorphism of {{math|''H''}}, then the product function {{math|''α'' × ''β'': ''G'' × ''H'' → ''G'' × ''H''}} defined by

:{{math|(''α'' × ''β'')(''g'', ''h'') {{=}} <big>(</big>''α''(''g''), ''β''(''h'')<big>)</big>}}

is an automorphism of {{math|''G'' × ''H''}}.  It follows that {{math|Aut(''G'' × ''H'')}} has a subgroup isomorphic
to the direct product {{math|Aut(''G'') × Aut(''H'')}}.

It is not true in general that every automorphism of {{math|''G'' × ''H''}} has the above form.  (That is, {{math|Aut(''G'') × Aut(''H'')}} is often a proper subgroup of {{math|Aut(''G'' × ''H'')}}.)  For example, if {{math|''G''}} is any group, then there exists an automorphism {{math|''σ''}} of {{math|''G'' × ''G''}} that switches the two factors, i.e.

:{{math|''σ''(''g''<sub>1</sub>, ''g''<sub>2</sub>) {{=}} (''g''<sub>2</sub>, ''g''<sub>1</sub>)}}.

For another example, the automorphism group of {{math|'''Z''' × '''Z'''}} is {{math|[[Modular group|''GL''(2, '''Z''')]]}}, the group of all {{math|2 × 2}} [[Matrix (mathematics)|matrices]] with integer entries and [[Determinant|determinant]], {{math|±1}}.  This automorphism group is infinite, but only finitely many of the automorphisms have the form given above.

In general, every [[Endomorphism|endomorphism]] of {{math|''G'' × ''H''}} can be written as a {{math|2 × 2}} matrix

:<math>\begin{bmatrix}\alpha & \beta \\ \gamma & \delta\end{bmatrix}</math>

where {{math|''α''}} is an endomorphism of {{math|''G''}}, {{math|''δ''}} is an endomorphism of {{math|''H''}}, and {{math| ''β'': ''H'' → ''G''}} and {{math| ''γ'': ''G'' → ''H''}} are homomorphisms.  Such a matrix must have the property that every element in the [[Image (mathematics)|image]] of {{math|''α''}} commutes with every element in the image of {{math|''β''}}, and every element in the image of {{math|''γ''}} commutes with every element in the image of {{math|''δ''}}.

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|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 {{math|''G''<sub>1</sub>, ..., ''G''<sub>''n''</sub>}} of groups, the '''direct product'''

:<math>\prod_{i=1}^n G_i \;=\; G_1 \times G_2 \times \cdots \times G_n</math>
is defined as follows:{{unordered list
| The elements of {{math|''G''<sub>1</sub> × ⋯ × ''G''<sub>''n''</sub>}} are [[Tuple|tuple]]s {{math|(''g''<sub>1</sub>, ..., ''g''<sub>''n''</sub>)}}, where {{math|''g<sub>i</sub>'' ∈ ''G<sub>i</sub>''}} for each {{math|''i''}}.
| The operation on {{math|''G''<sub>1</sub> × ⋯ × ''G''<sub>''n''</sub>}} is defined component-wise:
:{{math|(''g''<sub>1</sub>, ..., ''g<sub>n</sub>'')(''g''<sub>1</sub>′, ..., ''g<sub>n</sub>''′)}} {{=}} {{math|(''g''<sub>1</sub>''g''<sub>1</sub>′, ..., ''g<sub>n</sub>g<sub>n</sub>''′)}}.
}}

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 {{math|''G''<sub>1</sub>, ''G''<sub>2</sub>, ...}} 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|indexed family]] {&nbsp;{{math|''G<sub>i</sub>''}}&nbsp;}{{math|<sub>''i''∈''I''</sub>}} of groups, the '''direct product''' {{math|Π<sub>''i''∈''I''</sub> ''G<sub>i</sub>''}} is defined as follows:{{unordered list
| The elements of {{math|Π<sub>''i''∈''I''</sub> ''G<sub>i</sub>''}} are the elements of the [[Cartesian product#Infinite products|infinite Cartesian product]] of the sets {{math|''G<sub>i</sub>''}}; i.e., functions {{math|ƒ: ''I'' → ⋃<sub>''i''∈''I''</sub> ''G<sub>i</sub>''}} with the property that {{math|ƒ(''i'') ∈ ''G<sub>i</sub>''}} for each&nbsp;{{math|''i''}}.
| The product of two elements {{math|ƒ, ''g''}} is defined componentwise:
:{{math|(ƒ • ''g'')(''i'')}} {{=}} {{math|ƒ(''i'') • ''g''(''i'')}}.
}}

Unlike a finite direct product, the infinite direct product {{math|Π<sub>''i''∈''I''</sub> ''G<sub>i</sub>''}} is not generated by the elements of the isomorphic subgroups {&nbsp;{{math|''G<sub>i</sub>''}}&nbsp;}{{math|<sub>''i''∈''I''</sub>}}.  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====
{{main|Semidirect product}}
Recall that a group {{math|''P''}} with subgroups {{math|''G''}} and {{math|''H''}} is isomorphic to the direct product of {{math|''G''}} and {{math|''H''}} as long as it satisfies the following three conditions:
# The [[Intersection (set theory)|intersection]] {{math|''G''  ∩  ''H''}} is [[Trivial group|trivial]].
# Every element of {{math|''P''}} can be expressed uniquely as the product of an element of {{math|''G''}} and an element of&nbsp;{{math|''H''}}.
# Both {{math|''G''}} and {{math|''H''}} are [[Normal subgroup|normal]] in {{math|''P''}}.
A '''semidirect product''' of {{math|''G''}} and {{math|''H''}} is obtained by relaxing the third condition, so that only one of the two subgroups {{math|''G'', ''H''}} is required to be normal.  The resulting product still consists of ordered pairs {{math|(''g'', ''h'')}}, 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 {{math|''P''}} is referred to as a '''[[Zappa–Szép product]]''' of {{math|''G''}} and {{math|''H''}}.

====Free products====
{{main|Free product}}
The '''free product''' of {{math|''G''}} and {{math|''H''}}, usually denoted {{math|''G'' ∗ ''H''}}, is similar to the direct product, except that the subgroups {{math|''G''}} and {{math|''H''}} of {{math|''G'' ∗ ''H''}} are not required to commute.  That is, if

:{{math|''G''}} = {{math|〈 ''S<sub>G</sub>'' }}<big>|</big>{{math| ''R<sub>G</sub>'' 〉}} &nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp; {{math|''H''}} = {{math|〈 ''S<sub>H</sub>'' }}<big>|</big>{{math| ''R<sub>H</sub>'' 〉}},

are presentations for {{math|''G''}} and {{math|''H''}}, then

:{{math|''G'' ∗ ''H''}} = {{math|〈 ''S<sub>G</sub>'' ∪ ''S<sub>H</sub>'' }}<big>|</big>{{math| ''R<sub>G</sub>'' ∪ ''R<sub>H</sub>'' 〉}}.

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|coproduct]] in the [[Category of groups|category of groups]].

====Subdirect products====
{{main|Subdirect product}}
If {{math|''G''}} and {{math|''H''}} are groups, a '''subdirect product''' of {{math|''G''}} and {{math|''H''}} is any subgroup of {{math|''G'' × ''H''}} which maps [[Surjective function|surjectively]] onto {{math|''G''}} and {{math|''H''}} under the projection homomorphisms. By [[Goursat's lemma]], every subdirect product is a fiber product.

====Fiber products====
{{main|Pullback (category theory)}}
Let {{math|''G''}}, {{math|''H''}}, and {{math|''Q''}} be groups, and let {{math|''φ'': ''G'' → ''Q''}} and {{math|''χ'': ''H'' → ''Q''}} be homomorphisms.  The '''fiber product''' of {{math|''G''}} and {{math|''H''}} over {{math|''Q''}}, also known as a [[Pullback (category theory)|pullback]], is the following subgroup of {{math|''G'' × ''H''}}:

: {{math|''G'' ×<sub>''Q''</sub> ''H''}} &nbsp;=&nbsp; <big>{</big> {{math|(''g'', ''h'') ∈ ''G'' × ''H'' : ''φ(g)'' {{=}} ''χ(h)''}} <big>}</big>.
If {{math|''φ'': ''G'' → ''Q''}} and {{math|''χ'': ''H'' → ''Q''}} are [[Epimorphism|epimorphism]]s, then this is a subdirect product.

==References==
<references />
* {{Citation
  | last1=Artin 
  | first1=Michael
  | title=Algebra
  | publisher=Prentice Hall
  | isbn=978-0-89871-510-1
  | year=1991
 }}
* {{Citation | last1=Herstein | first1=Israel Nathan | title=Abstract algebra | publisher=Prentice Hall Inc. | location=Upper Saddle River, NJ | edition=3rd | isbn=978-0-13-374562-7 | mr=1375019 | year=1996}}.
* {{Citation | last1=Herstein | first1=Israel Nathan | title=Topics in algebra | publisher=Xerox College Publishing | location=Lexington, Mass. | edition=2nd | mr=0356988 | year=1975}}.
* {{Lang Algebra}}<!-- Don't add a fullstop here: it breaks the layout! -->
* {{Citation | last1=Lang | first1=Serge | title=Undergraduate Algebra | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | location=Berlin, New York | edition=3rd | isbn=978-0-387-22025-3 | year=2005}}.
* {{Citation | last1=Robinson | first1=Derek John Scott | title=A course in the theory of groups | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-94461-6 | year=1996}}.

[[Category:Group theory]]

{{Sourceattribution|Direct product of groups|1}}