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In mathematics, given a [[group (mathematics)|group]] ''G'' under a [[binary operation]]&nbsp;∗, a [[subset]] ''H'' of ''G'' is called a '''subgroup''' of ''G'' if ''H'' also forms a group under the operation&nbsp;∗. More precisely, ''H'' is a subgroup of ''G'' if the [[function (mathematics)#Restrictions and extensions|restriction]] of ∗ to {{nowrap|''H'' × ''H''}} is a group operation on ''H''. This is usually represented notationally by {{nowrap|''H'' ≤ ''G''}}, read as "''H'' is a subgroup of ''G''".

A '''proper subgroup''' of a group ''G'' is a subgroup ''H'' which is a [[subset|proper subset]] of ''G'' (i.e. {{nowrap|''H'' ≠ ''G''}}). The '''trivial subgroup''' of any group is the subgroup {''e''} consisting of just the identity element. If ''H'' is a subgroup of ''G'', then ''G'' is sometimes called an ''overgroup'' of ''H''.

The same definitions apply more generally when ''G'' is an arbitrary [[semigroup]], but this article will only deal with subgroups of groups. The group ''G'' is sometimes denoted by the ordered pair {{nowrap|(''G'', ∗)}}, usually to emphasize the operation ∗ when ''G'' carries multiple algebraic or other structures.

This article will write ''ab'' for {{nowrap|''a'' ∗ ''b''}}, as is usual.

==Basic properties of subgroups==

*A subset ''H'' of the group ''G'' is a subgroup of ''G'' if and only if it is nonempty and closed under products and inverses.  (The closure conditions mean the following: whenever ''a'' and ''b'' are in ''H'', then ''ab'' and ''a''<sup>&minus;1</sup> are also in ''H''. These two conditions can be combined into one equivalent condition: whenever ''a'' and ''b'' are in ''H'', then ''ab''<sup>&minus;1</sup> is also in ''H''.)  In the case that ''H'' is finite, then ''H'' is a subgroup [[if and only if]] ''H'' is closed under products. (In this case, every element ''a'' of ''H'' generates a finite cyclic subgroup of ''H'', and the inverse of ''a'' is then ''a''<sup>&minus;1</sup> = ''a''<sup>''n'' &minus; 1</sup>, where ''n'' is the order of ''a''.)
*The above condition can be stated in terms of a [[homomorphism]]; that is, ''H'' is a subgroup of a group ''G'' if and only if ''H'' is a subset of ''G'' and there is an inclusion homomorphism (i.e., i(''a'') = ''a'' for every ''a'') from ''H'' to ''G''.
*The [[Identity element|identity]] of a subgroup is the identity of the group: if ''G'' is a group with identity ''e''<sub>''G''</sub>, and ''H'' is a subgroup of ''G'' with identity ''e''<sub>''H''</sub>, then ''e''<sub>''H''</sub> = ''e''<sub>''G''</sub>.
*The [[Inverse element|inverse]] of an element in a subgroup is the inverse of the element in the group: if ''H'' is a subgroup of a group ''G'', and ''a'' and ''b'' are elements of ''H'' such that ''ab'' = ''ba'' = ''e''<sub>''H''</sub>, then ''ab'' = ''ba'' = ''e''<sub>''G''</sub>.
*The [[Intersection (set theory)|intersection]] of subgroups ''A'' and ''B'' is again a subgroup.<ref>Jacobson (2009), p. 41</ref> The [[Union (set theory)|union]] of subgroups ''A'' and ''B'' is a subgroup if and only if either ''A'' or ''B'' contains the other, since for example 2 and 3 are  in the union of 2Z and 3Z but their sum 5 is not. Another example is the union of the x-axis and the y-axis in the plane (with the addition operation); each of these objects is a subgroup but their union is not. This also serves as an example of two subgroups, whose intersection is precisely the identity.
*If ''S'' is a subset of ''G'', then there exists a minimum subgroup containing ''S'', which can be found by taking the intersection of all of subgroups containing ''S''; it is denoted by <''S''> and is said to be the [[generating set of a group|subgroup generated by ''S'']]. An element of ''G'' is in <''S''> if and only if it is a finite product of elements of ''S'' and their inverses.
*Every element ''a'' of a group ''G'' generates the cyclic subgroup <''a''>. If <''a''> is [[group isomorphism|isomorphic]] to '''Z'''/''n'''''Z''' for some positive integer ''n'', then ''n'' is the smallest positive integer for which ''a''<sup>''n''</sup> = ''e'', and ''n'' is called the ''order'' of ''a''. If <''a''> is isomorphic to '''Z''', then ''a'' is said to have ''infinite order''.
*The subgroups of any given group form a [[complete lattice]] under inclusion, called the [[lattice of subgroups]]. (While the [[infimum]] here is the usual set-theoretic intersection, the [[supremum]] of a set of subgroups is the subgroup ''generated by'' the set-theoretic union of the subgroups, not the set-theoretic union itself.) If ''e'' is the identity of ''G'', then the trivial group {''e''} is the [[partial order|minimum]] subgroup of ''G'', while the [[partial order|maximum]] subgroup is the group ''G'' itself.

[[File:Left cosets of Z 2 in Z 8.svg|thumb|G is the group <math>\mathbb{Z}/8\mathbb{Z}</math>, the [[Integers_modulo_n|integers mod 8]] under addition. The subgroup H contains only 0 and 4, and is isomorphic to <math>\mathbb{Z}/2\mathbb{Z}</math>. There are four left cosets of H: H itself, 1+H, 2+H, and 3+H (written using additive notation since this is an [[Abelian group|additive group]]). Together they partition the entire group G into equal-size, non-overlapping sets. The index [G : H] is 4.]]
==Cosets and Lagrange's theorem==
Given a subgroup ''H'' and some ''a'' in G, we define the '''left [[coset]]''' ''aH'' = {''ah'' : ''h'' in ''H''}. Because ''a'' is invertible, the map φ : ''H'' → ''aH''  given by φ(''h'') = ''ah'' is a [[bijection]]. Furthermore, every element of ''G'' is contained in precisely one left coset of ''H''; the left cosets are the equivalence classes corresponding to the [[equivalence relation]] ''a''<sub>1</sub> ~ ''a''<sub>2</sub> [[if and only if]] ''a''<sub>1</sub><sup>−1</sup>''a''<sub>2</sub> is in ''H''. The number of left cosets of ''H'' is called the [[Index of a subgroup|index]] of ''H'' in ''G'' and is denoted by [''G'' : ''H'']. 

[[Lagrange's theorem (group theory)|Lagrange's theorem]] states that for a finite group ''G'' and a subgroup ''H'', 
:<math> [ G : H ] = { |G| \over |H| } </math>
where |''G''| and  |''H''| denote the [[order (group theory)|order]]s of ''G'' and ''H'', respectively. In particular, the order of every subgroup of ''G'' (and the order of every element of ''G'') must be a [[divisor]] of  |''G''|.

'''Right cosets''' are defined analogously: ''Ha'' = {''ha'' : ''h'' in ''H''}. They are also the equivalence classes for a suitable equivalence relation and their number is equal to [''G'' : ''H''].  

If ''aH'' = ''Ha'' for every ''a'' in ''G'', then ''H'' is said to be a [[normal subgroup]]. Every subgroup of index 2 is normal: the left cosets, and also the right cosets, are simply the subgroup and its complement. More generally, if ''p'' is the lowest prime dividing the order of a finite group ''G,'' then any subgroup of index ''p'' (if such exists) is normal.

==Example: Subgroups of Z<sub>8</sub>==<!-- This section is linked from [[List of small groups]] -->
Let ''G'' be the [[cyclic group]] Z<sub>8</sub> whose elements are
:<math>G=\left\{0,2,4,6,1,3,5,7\right\}</math>
and whose group operation is [[modular arithmetic|addition modulo eight]].  Its [[Cayley table]] is
{| border="2" cellpadding="7"
!style="background:#FFFFAA;"| +
!style="background:#FFFFAA;"| <span style="color:red;">0</span>
!style="background:#FFFFAA;"| <span style="color:red;">2</span>
!style="background:#FFFFAA;"| <span style="color:red;">4</span>
!style="background:#FFFFAA;"| <span style="color:red;">6</span>
!style="background:#FFFFAA;"| <span style="color:blue;">1</span>
!style="background:#FFFFAA;"| <span style="color:blue;">3</span>
!style="background:#FFFFAA;"| <span style="color:blue;">5</span>
!style="background:#FFFFAA;"| <span style="color:blue;">7</span>
|-
!style="background:#FFFFAA;"| <span style="color:red;">0</span>
| <span style="color:orange;">0</span> || <span style="color:red;">2</span> || <span style="color:orange;">4</span> || <span style="color:red;">6</span> || <span style="color:blue;">1</span> || <span style="color:blue;">3</span> || <span style="color:blue;">5</span> || <span style="color:blue;">7</span>
|-
!style="background:#FFFFAA;"| <span style="color:red;">2</span>
| <span style="color:red;">2</span> || <span style="color:red;">4</span> || <span style="color:red;">6</span> || <span style="color:red;">0</span> || <span style="color:blue;">3</span> || <span style="color:blue;">5</span> || <span style="color:blue;">7</span> || <span style="color:blue;">1</span>
|-
!style="background:#FFFFAA;"| <span style="color:red;">4</span>
| <span style="color:orange;">4</span> || <span style="color:red;">6</span> || <span style="color:orange;">0</span> || <span style="color:red;">2</span> || <span style="color:blue;">5</span> || <span style="color:blue;">7</span> || <span style="color:blue;">1</span> || <span style="color:blue;">3</span>
|-
!style="background:#FFFFAA;"| <span style="color:red;">6</span>
| <span style="color:red;">6</span> || <span style="color:red;">0</span> || <span style="color:red;">2</span> || <span style="color:red;">4</span> || <span style="color:blue;">7</span> || <span style="color:blue;">1</span> || <span style="color:blue;">3</span> || <span style="color:blue;">5</span>
|-
!style="background:#FFFFAA;"| <span style="color:blue;">1</span>
| <span style="color:blue;">1</span> || <span style="color:blue;">3</span> || <span style="color:blue;">5</span> || <span style="color:blue;">7</span> || <span style="color:red;">2</span> || <span style="color:red;">4</span> || <span style="color:red;">6</span> || <span style="color:red;">0</span>
|-
!style="background:#FFFFAA;"| <span style="color:blue;">3</span>
| <span style="color:blue;">3</span> || <span style="color:blue;">5</span> || <span style="color:blue;">7</span> || <span style="color:blue;">1</span> || <span style="color:red;">4</span> || <span style="color:red;">6</span> || <span style="color:red;">0</span> || <span style="color:red;">2</span>
|-
!style="background:#FFFFAA;"| <span style="color:blue;">5</span>
| <span style="color:blue;">5</span> || <span style="color:blue;">7</span> || <span style="color:blue;">1</span> || <span style="color:blue;">3</span> || <span style="color:red;">6</span> || <span style="color:red;">0</span> || <span style="color:red;">2</span> || <span style="color:red;">4</span>
|-
!style="background:#FFFFAA;"| <span style="color:blue;">7</span>
| <span style="color:blue;">7</span> || <span style="color:blue;">1</span> || <span style="color:blue;">3</span> || <span style="color:blue;">5</span> || <span style="color:red;">0</span> || <span style="color:red;">2</span> || <span style="color:red;">4</span> || <span style="color:red;">6</span>
|}

This group has two nontrivial subgroups: <span style="color:orange;">''J''={0,4}</span> and <span style="color:red;">''H''={0,2,4,6}</span>, where ''J'' is also a subgroup of ''H''.  The Cayley table for ''H'' is the top-left quadrant of the Cayley table for ''G''.  The group ''G'' is [[cyclic group|cyclic]], and so are its subgroups.  In general, subgroups of cyclic groups are also cyclic.

==Example: Subgroups of S<sub>4 </sub>(the [[symmetric group]] on 4 elements)==
Every group has as many small subgroups as neutral elements on the main diagonal:

The [[w:trivial group|trivial group]] and two-element groups Z<sub>2</sub>. These small subgroups are not counted in the following list.
{| style="width:100%"
| style="vertical-align:top;"|[[File:Symmetric group 4; Cayley table; numbers.svg|thumb|left|595px|The [[symmetric group]] S<sub>4</sub> showing all [[permutation]]s of 4 elements]]
| style="vertical-align:top;"|[[File:Symmetric group 4; Lattice of subgroups Hasse diagram.svg|thumb|right|[[Hasse diagram]] of the [[lattice of subgroups]] of S<sub>4</sub>]]
|}

===12 elements===
[[File:Alternating group 4; Cayley table; numbers.svg|thumb|left|323px|The [[w:Alternating group|alternating group]] A<sub>4</sub> showing only the [[w:parity of a permutation|even permutations]]<br><br>Subgroups:<br>[[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg|70px]]<br>[[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,3,4).svg|60px]][[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,11,19).svg|60px]]  [[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,15,20).svg|60px]]  [[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,8,12).svg|60px]]]]
<br clear=all>

===8 elements===
{| 
|-
| <!-- LEFT -->[[File:Dihedral group of order 8; Cayley table (element orders 1,2,2,2,2,4,4,2); subgroup of S4.svg|thumb|233px|[[w:Dihedral group|Dihedral group]] [[Dihedral group of order 8|of order 8]]<br><br>Subgroups:<br>[[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,1,6,7).svg|70px]][[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg|70px]][[File:Cyclic group 4; Cayley table (element orders 1,2,4,4); subgroup of S4.svg|70px]]]] || &nbsp; || <!-- CENTRAL -->[[File:Dihedral group of order 8; Cayley table (element orders 1,2,2,4,2,2,4,2); subgroup of S4.svg|thumb|233px|Dihedral group of order 8<br><br>Subgroups:<br>[[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,5,14,16).svg|70px]][[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg|70px]][[File:Cyclic group 4; Cayley table (element orders 1,4,2,4); subgroup of S4.svg|70px]]]] || &nbsp; || <!-- RIGHT -->[[File:Dihedral group of order 8; Cayley table (element orders 1,2,2,4,4,2,2,2); subgroup of S4.svg|thumb|233px|Dihedral group of order 8<br><br>Subgroups:<br>[[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,2,21,23).svg|70px]][[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg|70px]][[File:Cyclic group 4; Cayley table (element orders 1,4,4,2); subgroup of S4.svg|70px]]]]
|}
<br clear=all>

===6 elements===
{| 
|-
| <!-- 1 -->[[File:Symmetric group 3; Cayley table; subgroup of S4 (elements 0,1,2,3,4,5).svg|thumb|187px|[[w:Symmetric group|Symmetric group]] [[w:Dihedral group of order 6|S<sub>3</sub>]]<br><br>Subgroup:[[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,3,4).svg|60px]]]] ||  <!-- 2 -->[[File:Symmetric group 3; Cayley table; subgroup of S4 (elements 0,5,6,11,19,21).svg|thumb|187px|Symmetric group S<sub>3</sub><br><br>Subgroup:[[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,11,19).svg|60px]]]]  || <!-- 3 -->[[File:Symmetric group 3; Cayley table; subgroup of S4 (elements 0,1,14,15,20,21).svg|thumb|187px|Symmetric group S<sub>3</sub><br><br>Subgroup:[[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,15,20).svg|60px]]]] ||  <!-- 4 -->[[File:Symmetric group 3; Cayley table; subgroup of S4 (elements 0,2,6,8,12,14).svg|thumb|187px|Symmetric group S<sub>3</sub><br><br>Subgroup:[[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,8,12).svg|60px]]]]
|}
<br clear=all>

===4 elements===
{|
|-
| [[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,1,6,7).svg|thumb|142px|[[w:Klein four-group|Klein four-group]]]] || [[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,5,14,16).svg|thumb|142px|Klein four-group]] || [[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,2,21,23).svg|thumb|142px|Klein four-group]] || [[File:Klein four-group; Cayley table; subgroup of S4 (elements 0,7,16,23).svg|thumb|142px|Klein four-group]]
|}
<br clear=all>
{|
|-
| [[File:Cyclic group 4; Cayley table (element orders 1,2,4,4); subgroup of S4.svg|thumb|142px|[[w:Cyclic group|Cyclic group]] Z<sub>4</sub>]] || [[File:Cyclic group 4; Cayley table (element orders 1,4,2,4); subgroup of S4.svg|thumb|142px|Cyclic group Z<sub>4</sub>]] || [[File:Cyclic group 4; Cayley table (element orders 1,4,4,2); subgroup of S4.svg|thumb|142px|Cyclic group Z<sub>4</sub>]]
|}
<br clear=all>

===3 elements===
{|
|-
| [[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,3,4).svg|thumb|120px|[[w:Cyclic group|Cyclic group]] Z<sub>3</sub>]] || [[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,11,19).svg|thumb|120px|Cyclic group Z<sub>3</sub>]] || [[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,15,20).svg|thumb|120px|Cyclic group Z<sub>3</sub>]] || 
[[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,8,12).svg|thumb|120px|Cyclic group Z<sub>3</sub>]]
|}
<br clear=all>

== See also ==
* [[Cartan subgroup]]
* [[Fitting subgroup]]
* [[Stable subgroup]]

== Notes ==
<references/>

== References ==
* {{Citation| last=Jacobson| first=Nathan| author-link=Nathan Jacobson| year=2009| title=Basic algebra| edition=2nd| volume = 1 | series= | publisher=Dover| isbn = 978-0-486-47189-1}}.

[[Category:Group theory]]
[[Category:Subgroup properties]]