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| | Hello. Let me introduce the author. Her name is Refugia Shryock. Doing ceramics is what her family and her appreciate. Years in the past he moved to North Dakota and his family loves it. She is a librarian but she's always needed her own company.<br><br>my web blog - [http://bit.ly/1pABYYJ weight loss food delivery] |
| {{About|the mathematical concept|the galaxy-related concept|galaxy group}}
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| {{Group theory sidebar |Basics}}
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| In mathematics, given a [[group (mathematics)|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 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''".
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| 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''.
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| 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.
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| This article will write ''ab'' for {{nowrap|''a'' ∗ ''b''}}, as is usual.
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| ==Basic properties of subgroups==
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| *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>−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>−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>−1</sup> = ''a''<sup>''n'' − 1</sup>, where ''n'' is the order of ''a''.)
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| *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''.
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| *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>.
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| *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>.
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| *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.
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| *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.
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| *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''.
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| *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.
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| [[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.]]
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| ==Cosets and Lagrange's theorem==
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| 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''].
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| [[Lagrange's theorem (group theory)|Lagrange's theorem]] states that for a finite group ''G'' and a subgroup ''H'',
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| :<math> [ G : H ] = { |G| \over |H| } </math>
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| 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''|.
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| '''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''].
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| 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.
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| ==Example: Subgroups of Z<sub>8</sub>==<!-- This section is linked from [[List of small groups]] -->
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| Let ''G'' be the [[cyclic group]] Z<sub>8</sub> whose elements are
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| :<math>G=\left\{0,2,4,6,1,3,5,7\right\}</math>
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| and whose group operation is [[modular arithmetic|addition modulo eight]]. Its [[Cayley table]] is
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| {| border="2" cellpadding="7"
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| !style="background:#FFFFAA;"| +
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| !style="background:#FFFFAA;"| <span style="color:red;">0</span>
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| !style="background:#FFFFAA;"| <span style="color:red;">2</span>
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| !style="background:#FFFFAA;"| <span style="color:red;">4</span>
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| !style="background:#FFFFAA;"| <span style="color:red;">6</span>
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| !style="background:#FFFFAA;"| <span style="color:blue;">1</span>
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| !style="background:#FFFFAA;"| <span style="color:blue;">3</span>
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| !style="background:#FFFFAA;"| <span style="color:blue;">5</span>
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| !style="background:#FFFFAA;"| <span style="color:blue;">7</span>
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| |-
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| !style="background:#FFFFAA;"| <span style="color:red;">0</span>
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| | <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>
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| |-
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| !style="background:#FFFFAA;"| <span style="color:red;">2</span>
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| | <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>
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| |-
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| !style="background:#FFFFAA;"| <span style="color:red;">4</span>
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| | <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>
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| |-
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| !style="background:#FFFFAA;"| <span style="color:red;">6</span>
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| | <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>
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| |-
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| !style="background:#FFFFAA;"| <span style="color:blue;">1</span>
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| | <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>
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| |-
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| !style="background:#FFFFAA;"| <span style="color:blue;">3</span>
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| | <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>
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| |-
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| !style="background:#FFFFAA;"| <span style="color:blue;">5</span>
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| | <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>
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| |-
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| !style="background:#FFFFAA;"| <span style="color:blue;">7</span>
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| | <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>
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| |}
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| 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.
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| ==Example: Subgroups of S<sub>4 </sub>(the [[symmetric group]] on 4 elements)==
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| Every group has as many small subgroups as neutral elements on the main diagonal:
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| The [[w:trivial group|trivial group]] and two-element groups Z<sub>2</sub>. These small subgroups are not counted in the following list.
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| {| style="width:100%"
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| | 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]]
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| | 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>]]
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| |}
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| ===12 elements===
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| [[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]]]]
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| <br clear=all>
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| ===8 elements===
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| {|
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| | <!-- 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]]]] || || <!-- 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]]]] || || <!-- 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]]]]
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| |}
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| <br clear=all>
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| ===6 elements===
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| {|
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| | <!-- 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]]]]
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| |}
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| <br clear=all>
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| ===4 elements===
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| {|
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| | [[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]]
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| |}
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| <br clear=all>
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| {|
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| | [[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>]]
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| |}
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| <br clear=all>
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| ===3 elements===
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| {|
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| | [[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>]] ||
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| [[File:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,8,12).svg|thumb|120px|Cyclic group Z<sub>3</sub>]]
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| |}
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| <br clear=all>
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| == See also ==
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| * [[Cartan subgroup]]
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| * [[Fitting subgroup]]
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| * [[Stable subgroup]]
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| == Notes ==
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| <references/>
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| == References ==
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| * {{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}}.
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| [[Category:Group theory]]
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| [[Category:Subgroup properties]]
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