View source for Subgroup

{{short description|Subset of a group that forms a group itself}}
{{other uses}}
{{Group theory sidebar |Basics}}

In [[group theory]], a branch of [[mathematics]], a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. 

Formally, given a [[group (mathematics)|group]] {{mvar|G}} under a [[binary operation]]&nbsp;∗, a [[subset]] {{mvar|H}} of {{mvar|G}} is called a '''subgroup''' of {{mvar|G}} if {{mvar|H}} also forms a group under the operation&nbsp;∗. More precisely, {{mvar|H}} is a subgroup of {{mvar|G}} if the [[Restriction (mathematics)|restriction]] of ∗ to {{math|''H'' × ''H''}} is a group operation on {{mvar|H}}. This is often denoted {{math|''H'' ≤ ''G''}}, read as "{{mvar|H}} is a subgroup of {{mvar|G}}".

The '''trivial subgroup''' of any group is the subgroup {''e''} consisting of just the identity element.{{sfn|Gallian|2013|p=61}}

A '''proper subgroup''' of a group {{mvar|G}} is a subgroup {{mvar|H}} which is a [[subset|proper subset]] of {{mvar|G}} (that is, {{math|''H'' ≠ ''G''}}). This is often represented notationally by {{math|''H'' < ''G''}}, read as "{{mvar|H}} is a proper subgroup of {{mvar|G}}". Some authors also exclude the trivial group from being proper (that is, {{math|''H'' ≠ {''e''}{{0ws}}}}).{{sfn|Hungerford|1974|p=32}}{{sfn|Artin|2011|p=43}}

If {{mvar|H}} is a subgroup of {{mvar|G}}, then {{mvar|G}} is sometimes called an '''overgroup''' of {{mvar|H}}.

The same definitions apply more generally when {{mvar|G}} is an arbitrary [[semigroup]], but this article will only deal with subgroups of groups.

==Subgroup tests==

Suppose that {{mvar|G}} is a group, and {{mvar|H}} is a subset of {{mvar|G}}.  For now, assume that the group operation of {{mvar|G}} is written multiplicatively, denoted by juxtaposition.
*Then {{mvar|H}} is a subgroup of {{mvar|G}} [[if and only if]] {{mvar|H}} is nonempty and [[Closure (mathematics)|closed]] under products and inverses. ''Closed under products'' means that for every {{mvar|a}} and {{mvar|b}} in {{mvar|H}}, the product {{mvar|ab}} is in {{mvar|H}}.  ''Closed under inverses'' means that for every {{mvar|a}} in {{mvar|H}}, the inverse {{math|''a''<sup>&minus;1</sup>}} is in {{mvar|H}}. These two conditions can be combined into one, that for every {{mvar|a}} and {{mvar|b}} in {{mvar|H}}, the element {{math|''ab''<sup>&minus;1</sup>}} is in {{mvar|H}}, but it is more natural and usually just as easy to test the two closure conditions separately.{{sfn|Kurzweil|Stellmacher|1998|p=4}}
*When {{mvar|H}} is ''finite'', the test can be simplified: {{mvar|H}} is a subgroup if and only if it is nonempty and closed under products. These conditions alone imply that every element {{mvar|a}} of {{mvar|H}} generates a finite cyclic subgroup of {{mvar|H}}, say of order {{mvar|n}}, and then the inverse of {{mvar|a}} is {{math|''a''<sup>''n''&minus;1</sup>}}.{{sfn|Kurzweil|Stellmacher|1998|p=4}}
If the group operation is instead denoted by addition, then ''closed under products'' should be replaced by ''closed under addition'', which is the condition that for every {{mvar|a}} and {{mvar|b}} in {{mvar|H}}, the sum {{math|''a'' + ''b''}} is in {{mvar|H}}, and ''closed under inverses'' should be edited to say that for every {{mvar|a}} in {{mvar|H}}, the inverse {{math|−''a''}} is in {{mvar|H}}.

==Basic properties of subgroups==

*The [[Identity element|identity]] of a subgroup is the identity of the group: if {{mvar|G}} is a group with identity {{mvar|e<sub>G</sub>}}, and {{mvar|H}} is a subgroup of {{mvar|G}} with identity {{mvar|e<sub>H</sub>}}, then {{math|1=''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 {{mvar|H}} is a subgroup of a group {{mvar|G}}, and {{mvar|a}} and {{mvar|b}} are elements of {{mvar|H}} such that {{math|1=''ab'' = ''ba'' = ''e<sub>H</sub>''}}, then {{math|1=''ab'' = ''ba'' = ''e<sub>G</sub>''}}.
*If {{mvar|H}} is a subgroup of {{mvar|G}}, then the inclusion map {{math|''H'' → ''G''}} sending each element {{mvar|a}} of {{mvar|H}} to itself is a [[homomorphism]].
*The [[Intersection (set theory)|intersection]] of subgroups {{mvar|A}} and {{mvar|B}} of {{mvar|G}} is again a subgroup of {{mvar|G}}.{{sfn|Jacobson|2009|p=41}} For example, the intersection of the {{mvar|x}}-axis and {{mvar|y}}-axis in {{tmath|\R^2}} under addition is the trivial subgroup.  More generally, the intersection of an arbitrary collection of subgroups of {{mvar|G}} is a subgroup of {{mvar|G}}.
*The [[Union (set theory)|union]] of subgroups {{mvar|A}} and {{mvar|B}} is a subgroup if and only if {{math|''A'' ⊆ ''B''}} or {{math|''B'' ⊆ ''A''}}.  A non-example: {{tmath|2\Z \cup 3\Z}} is not a subgroup of {{tmath|\Z,}} because 2 and 3 are elements of this subset whose sum, 5, is not in the subset.   Similarly, the union of the {{mvar|x}}-axis and the {{mvar|y}}-axis in {{tmath|\R^2}} is not a subgroup of {{tmath|\R^2.}}
*If {{mvar|S}} is a subset of {{mvar|G}}, then there exists a smallest subgroup containing {{mvar|S}}, namely the intersection of all of subgroups containing {{mvar|S}}; it is denoted by  {{math|{{angbr|''S''}}}} and is called the [[generating set of a group|subgroup generated by {{mvar|S}}]]. An element of {{mvar|G}} is in  {{math|{{angbr|''S''}}}} if and only if it is a finite product of elements of {{mvar|S}} and their inverses, possibly repeated.{{sfn|Ash|2002}}
*Every element {{mvar|a}} of a group {{mvar|G}} generates a cyclic subgroup {{math|{{angbr|''a''}}}}. If {{math|{{angbr|''a''}}}} is [[group isomorphism|isomorphic]] to {{tmath|\Z/n\Z}} ([[Integers modulo n|the integers {{math|mod ''n''}}]]) for some positive integer {{mvar|n}}, then {{mvar|n}} is the smallest positive integer for which {{math|1=''a<sup>n</sup>'' = ''e''}}, and {{mvar|n}} is called the ''order'' of {{mvar|a}}. If {{math|{{angbr|''a''}}}} is isomorphic to {{tmath|\Z,}} then {{mvar|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 {{mvar|e}} is the identity of {{mvar|G}}, then the trivial group {{math|{''e''} }} is the [[partial order|minimum]] subgroup of {{mvar|G}}, while the [[partial order|maximum]] subgroup is the group {{mvar|G}} itself.

[[File:Left cosets of Z 2 in Z 8.svg|thumb|{{mvar|G}} is the group <math>\Z/8\Z,</math> the [[Integers modulo n|integers mod 8]] under addition. The subgroup {{mvar|H}} contains only 0 and 4, and is isomorphic to <math>\Z/2\Z.</math> There are four left cosets of {{mvar|H}}: {{mvar|H}} itself, {{math|1 + ''H''}}, {{math|2 + ''H''}}, and {{math|3 + ''H''}} (written using additive notation since this is an [[Abelian group|additive group]]). Together they partition the entire group {{mvar|G}} into equal-size, non-overlapping sets. The index {{math|[''G'' : ''H'']}} is 4.]]

==Cosets and Lagrange's theorem==
{{Main|Coset|Lagrange's theorem (group theory)}}
Given a subgroup {{mvar|H}} and some {{mvar|a}} in {{mvar|G}}, we define the '''left [[coset]]''' {{math|1=''aH'' = {''ah'' : ''h'' in ''H''}.}} Because {{mvar|a}} is invertible, the map {{math|φ : ''H'' → ''aH''}}  given by {{math|1=φ(''h'') = ''ah''}} is a [[bijection]]. Furthermore, every element of {{mvar|G}} is contained in precisely one left coset of {{mvar|H}}; the left cosets are the equivalence classes corresponding to the [[equivalence relation]] {{math|''a''<sub>1</sub> ~ ''a''<sub>2</sub>}} [[if and only if]] {{tmath|a_1^{-1}a_2}} is in {{mvar|H}}. The number of left cosets of {{mvar|H}} is called the [[index of a subgroup|index]] of {{mvar|H}} in {{mvar|G}} and is denoted by {{math|[''G'' : ''H'']}}.

[[Lagrange's theorem (group theory)|Lagrange's theorem]] states that for a finite group {{mvar|G}} and a subgroup {{mvar|H}}, 
: <math> [ G : H ] = { |G| \over |H| }</math>
where {{mvar|{{abs|G}}}} and  {{mvar|{{abs|H}}}} denote the [[order (group theory)|order]]s of {{mvar|G}} and {{mvar|H}}, respectively. In particular, the order of every subgroup of {{mvar|G}} (and the order of every element of {{mvar|G}}) must be a [[divisor]] of {{mvar|{{abs|G}}}}.<ref>See a [https://www.youtube.com/watch?v=TCcSZEL_3CQ didactic proof in this video].</ref>{{sfn|Dummit|Foote|2004|p=90}}

'''Right cosets''' are defined analogously: {{math|1=''Ha'' = {''ha'' : ''h'' in ''H''}.}} They are also the equivalence classes for a suitable equivalence relation and their number is equal to {{math|[''G'' : ''H'']}}.

If {{math|1=''aH'' = ''Ha''}} for every {{mvar|a}} in {{mvar|G}}, then {{mvar|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 {{mvar|p}} is the lowest prime dividing the order of a finite group {{mvar|G}}, then any subgroup of index {{mvar|p}} (if such exists) is normal.

==Example: Subgroups of Z<sub>8</sub>==<!-- This section is linked from [[List of small groups]] -->
Let {{mvar|G}} be the [[cyclic group]] {{math|Z<sub>8</sub>}} whose elements are
:<math>G = \left\{0, 4, 2, 6, 1, 5, 3, 7\right\}</math>
and whose group operation is [[modular arithmetic|addition modulo 8]]. Its [[Cayley table]] is

{| class="wikitable" style="color:blue;"
|-
! style="background-color:#FFFFAA; color:black;" | +
! style="background-color:#FFFFAA; color:orange;" | 0
! style="background-color:#FFFFAA; color:orange;" | 4
! style="background-color:#FFFFAA; color:red;" | 2
! style="background-color:#FFFFAA; color:red;" | 6
! style="background-color:#FFFFAA;" | 1
! style="background-color:#FFFFAA;" | 5
! style="background-color:#FFFFAA;" | 3
! style="background-color:#FFFFAA;" | 7
|-
! style="background:#FFFFAA; color:orange;" | 0
| style="color:orange;" | 0 || style="color:orange;" | 4
| style="color:red;"    | 2 || style="color:red;"    | 6
| 1 || 5 || 3 || 7
|-
! style="background:#FFFFAA; color:orange;" | 4
| style="color:orange;" | 4 || style="color:orange;" | 0
| style="color:red;"    | 6 || style="color:red;"    | 2
| 5 || 1 || 7 || 3
|-
! style="background:#FFFFAA; color:red;" | 2
| style="color:red;" | 2 ||  style="color:red;" | 6 || style="color:red;" | 4 || style="color:red;" | 0
| 3 || 7 || 5 || 1
|-
! style="background:#FFFFAA; color:red;" | 6
| style="color:red;" | 6 || style="color:red;" | 2 || style="color:red;" | 0 || style="color:red;" | 4
| 7 || 3 || 1 || 5
|-
! style="background-color:#FFFFAA;" | 1
| 1 || 5 || 3 || 7 || 2 || 6 || 4 || 0
|-
! style="background-color:#FFFFAA;" | 5
| 5 || 1 || 7 || 3 || 6 || 2 || 0 || 4
|-
! style="background-color:#FFFFAA;" | 3
| 3 || 7 || 5 || 1 || 4 || 0 || 6 || 2
|-
! style="background-color:#FFFFAA;" | 7
| 7 || 3 || 1 || 5 || 0 || 4 || 2 || 6
|}

This group has two nontrivial subgroups: {{math|{{colorbull|orange}} ''J'' {{=}} {0, 4} }} and {{math|{{colorbull|red}} ''H'' {{=}} {0, 4, 2, 6} }}, where {{mvar|J}} is also a subgroup of {{mvar|H}}.  The Cayley table for {{mvar|H}} is the top-left quadrant of the Cayley table for {{mvar|G}}; The Cayley table for {{mvar|J}} is the top-left quadrant of the Cayley table for {{mvar|H}}.  The group {{mvar|G}} is [[cyclic group|cyclic]], and so are its subgroups.  In general, subgroups of cyclic groups are also cyclic.{{sfn|Gallian|2013|p=81}}

==Example: Subgroups of S<sub>4</sub>{{anchor|Subgroups of S4}}==
{{math|S<sub>4</sub>}} is the [[symmetric group]] whose elements correspond to the [[permutation]]s of 4 elements.<br>
Below are all its subgroups, ordered by cardinality.<br>
Each group <small>(except those of cardinality 1 and 2)</small> is represented by its [[Cayley table]].

===24 elements===
Like each group, {{math|S<sub>4</sub>}} is a subgroup of itself.

{| style="width:100%"
| style="vertical-align:top;"|[[File:Symmetric group 4; Cayley table; numbers.svg|thumb|left|595px|Symmetric group {{math|S<sub>4</sub>}}]]
| style="vertical-align:top;"|
{{multiple image
 | align = right
 | image1 = Symmetric group S4; lattice of subgroups Hasse diagram; all 30 subgroups.svg
 | width1 = 250
 | caption1 = All 30 subgroups
 | image2 = Symmetric group S4; lattice of subgroups Hasse diagram; 11 different cycle graphs.svg
 | width2 = 185
 | caption2 = Simplified
 | footer = [[Hasse diagram]]s of the [[lattice of subgroups]] of {{math|S<sub>4</sub>}}
}}
|}

===12 elements===
The [[alternating group]] contains only the [[w:parity of a permutation|even permutations]].<br>
It is one of the two nontrivial proper [[normal subgroup]]s of {{math|S<sub>4</sub>}}. <small>(The other one is its Klein subgroup.)</small>
[[File:Alternating group 4; Cayley table; numbers.svg|thumb|left|323px|Alternating group {{math|A<sub>4</sub>}}<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]]]]
{{clear}}

===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]]]]
|}
{{clear}}

===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]] {{math|[[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 {{math|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 {{math|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 {{math|S<sub>3</sub>}}<br><br>Subgroup:[[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,8,12).svg|60px]]]]
|}
{{clear}}

===4 elements===
{|
|- style="vertical-align: top;"
| [[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><small>([[normal subgroup]])</small>]]
|}
{{clear}}
{|
|-
| [[File:Cyclic group 4; Cayley table (element orders 1,2,4,4); subgroup of S4.svg|thumb|142px|[[w:Cyclic group|Cyclic group]] {{math|Z<sub>4</sub>}}]] || [[File:Cyclic group 4; Cayley table (element orders 1,4,2,4); subgroup of S4.svg|thumb|142px|Cyclic group {{math|Z<sub>4</sub>}}]] || [[File:Cyclic group 4; Cayley table (element orders 1,4,4,2); subgroup of S4.svg|thumb|142px|Cyclic group {{math|Z<sub>4</sub>}}]]
|}
{{clear}}

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

===2 elements===
Each permutation {{mvar|p}} of order 2 generates a subgroup {{math|{1, ''p''}}}.
These are the permutations that have only 2-cycles:<br>
* There are the 6 [[Cyclic permutation#Transpositions|transpositions]] with one 2-cycle. &nbsp; <small>(green background)</small>
* And 3 permutations with two 2-cycles. &nbsp; <small>(white background, bold numbers)</small>

===1 element===
The [[trivial group|trivial subgroup]] is the unique subgroup of order 1.

==Other examples==
*The even integers form a subgroup {{tmath|2\Z}} of the [[integer ring]] {{tmath|\Z:}} the sum of two even integers is even, and the negative of an even integer is even.
*An [[ideal (ring theory)#Definitions|ideal]] in a ring {{mvar|R}} is a subgroup of the additive group of {{mvar|R}}.
*A [[linear subspace]] of a [[vector space]] is a subgroup of the additive group of vectors.
*In an [[abelian group]], the elements of finite [[order (group theory)|order]] form a subgroup called the [[torsion subgroup]].

== See also ==
* [[Cartan subgroup]]
* [[Fitting subgroup]]
* [[Fixed-point subgroup]]
* [[Fully normalized subgroup]]
* [[Stable subgroup]]

== Notes ==
<references/>

== References ==
* {{Citation| last=Jacobson| first=Nathan | author-link=Nathan Jacobson| year=2009| title=Basic algebra| edition=2nd| volume = 1 | publisher=Dover| isbn = 978-0-486-47189-1}}.
* {{Citation| last=Hungerford| first=Thomas| author-link=Thomas W. Hungerford| year=1974| title=Algebra| edition=1st| publisher=Springer-Verlag| isbn =9780387905181}}.
* {{Citation| last=Artin| first=Michael| author-link=Michael Artin| year=2011| title=Algebra| edition=2nd| publisher=Prentice Hall| isbn = 9780132413770}}.
* {{Cite book|title=Abstract algebra|last1=Dummit|first1=David S.|last2=Foote|first2=Richard M.|date=2004|publisher=Wiley|isbn=9780471452348|edition=3rd|location=Hoboken, NJ|oclc=248917264}}
* {{Cite book |last=Gallian |first=Joseph A. | author-link=Joseph Gallian|title=Contemporary abstract algebra |date=2013 |publisher=Brooks/Cole Cengage Learning |isbn=978-1-133-59970-8 |edition=8th |location=Boston, MA |oclc=807255720}}
* {{Cite book|last1=Kurzweil|first1=Hans|last2=Stellmacher|first2=Bernd|date=1998|title=Theorie der endlichen Gruppen|url=http://dx.doi.org/10.1007/978-3-642-58816-7|series=Springer-Lehrbuch|doi=10.1007/978-3-642-58816-7|isbn=978-3-540-60331-3 }}
* {{Cite book |last=Ash |first=Robert B. |url=https://faculty.math.illinois.edu/~r-ash/Algebra.html |title=Abstract Algebra: The Basic Graduate Year |date=2002 |publisher=Department of Mathematics University of Illinois |language=en}}

{{Group navbox}}
[[Category:Group theory]]
[[Category:Subgroup properties]]