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The following list in [[mathematics]] contains the [[finite group]]s of small [[order (group theory)|order]] [[up to]] [[group isomorphism]].
Journalists and Other Writers  Elliott Janner from Marathon, has several hobbies and interests that include embroidery, health and fitness and church/church activities. Finds the charm in visiting destinations all over the planet, recently just coming back from Urnes Stave Church.<br><br>Feel free to visit my blog post :: [http://smartchoicecart.in/?author=5077 Http://Smartchoicecart.In]
 
The list can be used to determine which known group a given finite group ''G'' is isomorphic to: first determine the order of ''G'', then look up the candidates for that order in the list below. If you know whether ''G'' is abelian or not, some candidates can be eliminated right away. To distinguish between the remaining candidates, look at the orders of your group's elements, and match it with the orders of the candidate group's elements.
 
== Glossary ==
* Z<sub>''n''</sub>: the [[cyclic group]] of order ''n'' (the notation C<sub>''n''</sub> is also used; it is isomorphic to the [[additive group]] of '''Z'''/''n'''''Z''').
* Dih<sub>''n''</sub>: the [[dihedral group]] of order 2''n'' (often the notation D<sub>''n''</sub> or D<sub>2''n''</sub> is used )
* S<sub>''n''</sub>: the [[symmetric group]] of degree ''n'', containing the [[factorial|''n''!]] [[permutation]]s of ''n'' elements.
* A<sub>''n''</sub>: the [[alternating group]] of degree ''n'', containing the ''n''!/2 [[even permutation]]s of ''n'' elements.
* Dic<sub>''n''</sub>: the [[dicyclic group]] of order 4''n''.
 
The notations Z<sub>''n''</sub> and Dih<sub>''n''</sub> have the advantage that [[point groups in three dimensions]] C<sub>''n''</sub> and D<sub>''n''</sub> do not have the same notation. There are more [[isometry group]]s than these two, of the same abstract group type.
 
The notation {{nowrap|''G'' × ''H''}} stands for the [[direct product of groups|direct product]] of the two groups; ''G''<sup>''n''</sup> denotes the direct product of a group with itself ''n'' times. ''G'' <!--&#x22CA;--> <math>\rtimes</math> ''H'' stands for a [[semidirect product]] where ''H'' acts on ''G''; where the particular action of ''H'' on ''G'' is omitted, it is because all possible non-trivial actions result in the same product group, [[up to isomorphism]].
 
[[abelian group|Abelian]] and [[simple group]]s are noted. (For groups of order {{nowrap|''n'' < 60}}, the simple groups are precisely the cyclic groups Z<sub>''n''</sub>, for prime ''n''.) The equality sign ("=") denotes isomorphism.
 
The identity element in the [[cycle graph (algebra)|cycle graphs]] is represented by the black circle.  The lowest order for which the cycle graph does not uniquely represent a group is order 16.
 
In the lists of subgroups, the trivial group and the group itself are not listed. Where there are multiple isomorphic subgroups, their number is indicated in parentheses.
 
== List of small abelian groups ==
The finite abelian groups are either cyclic groups, or direct products thereof; see [[abelian group]]s.
 
{| class="wikitable"
|-----
! Order
! Group
! Subgroups
! Properties
! [[Cycle graph (algebra)|Cycle graph]]
|-----
! 1
| [[trivial group]], {{nowrap|1=Z<sub>1</sub> = S<sub>1</sub> = A<sub>2</sub>}}
| -
| various properties hold [[Trivial (mathematics)|trivially]]
| [[Image:GroupDiagramMiniC1.svg|center]]
|-----
! 2
| Z<sub>2</sub> = S<sub>2</sub> = Dih<sub>1</sub>
| -
| simple, the smallest non-trivial group
| [[Image:GroupDiagramMiniC2.svg|center]]
|-----
! 3
| Z<sub>3</sub> = A<sub>3</sub> || -
| simple
| [[Image:GroupDiagramMiniC3.svg|center]]
|-----
! rowspan="2" | 4
| Z<sub>4</sub> || Z<sub>2</sub> || &nbsp;&nbsp;
| [[Image:GroupDiagramMiniC4.svg|center]]
|-----
| [[Klein four-group]], {{nowrap|1={{SubSup|Z|2|2}} = Dih<sub>2</sub>}}
| Z<sub>2</sub> (3)|| the smallest non-cyclic group
| [[Image:GroupDiagramMiniD4.svg|center]]
|-----
! 5
| Z<sub>5</sub> || - || simple
| [[Image:GroupDiagramMiniC5.svg|center]]
|-----
! 6
| Z<sub>6</sub> = Z<sub>3</sub> × Z<sub>2</sub><ref>See a worked [[Isomorphism#Integers modulo 6|example showing the isomorphism Z<sub>6</sub> = Z<sub>3</sub> × Z<sub>2</sub>]].</ref>
| Z<sub>3</sub>, Z<sub>2</sub> || &nbsp;
| [[Image:GroupDiagramMiniC6.svg|center]]
|-----
! 7
| Z<sub>7</sub> || - || simple
| [[Image:GroupDiagramMiniC7.svg|center]]
|-----
! rowspan="3" | 8
| [[Subgroup#Example: Subgroups of Z8|Z<sub>8</sub>]] || Z<sub>4</sub>, Z<sub>2</sub>
| &nbsp;
| [[Image:GroupDiagramMiniC8.svg|center]]
|-----
| Z<sub>4</sub> × Z<sub>2</sub>
| {{SubSup|Z|2|2}}, Z<sub>4</sub> (2), Z<sub>2</sub> (3)
| &nbsp;
| [[Image:GroupDiagramMiniC2C4.svg|center]]
|-----
| {{SubSup|Z|2|3}}
| {{SubSup|Z|2|2}} (7), Z<sub>2</sub> (7)
| the non-identity elements correspond to the points in the [[Fano plane]], the {{nowrap|Z<sub>2</sub> × Z<sub>2</sub>}} subgroups to the lines
| [[Image:GroupDiagramMiniC2x3.svg|center]]
|-----
! rowspan="2" | 9
| Z<sub>9</sub>  
| Z<sub>3</sub>
| &nbsp;
| [[Image:GroupDiagramMiniC9.svg|center]]
|-----
| {{SubSup|Z|3|2}}
| Z<sub>3</sub> (4)
| &nbsp;
| [[Image:GroupDiagramMiniC3x2.svg|center]]
|-----
! 10
| Z<sub>10</sub> = Z<sub>5</sub> × Z<sub>2</sub>
| Z<sub>5</sub>, Z<sub>2</sub> || &nbsp;
| [[Image:GroupDiagramMiniC10.svg|center]]
|-----
! 11
| Z<sub>11</sub> || - || simple
| [[Image:GroupDiagramMiniC11.svg|center]]
|-----
! rowspan="2" | 12
| Z<sub>12</sub> = Z<sub>4</sub> × Z<sub>3</sub>
| Z<sub>6</sub>, Z<sub>4</sub>, Z<sub>3</sub>, Z<sub>2</sub>
| &nbsp;
| [[Image:GroupDiagramMiniC12.svg|center]]
|-----
| Z<sub>6</sub> × Z<sub>2</sub> = Z<sub>3</sub> × {{SubSup|Z|2|2}}
| Z<sub>6</sub> (3), Z<sub>3</sub>, Z<sub>2</sub> (3), {{SubSup|Z|2|2}}
| &nbsp;
| [[Image:GroupDiagramMiniC2C6.svg|center]]
|-----
! 13
| Z<sub>13</sub> || - || simple
| [[Image:GroupDiagramMiniC13.svg|center]]
|-----
! 14
| Z<sub>14</sub> = Z<sub>7</sub> × Z<sub>2</sub>
| Z<sub>7</sub>, Z<sub>2</sub> || &nbsp;
| [[Image:GroupDiagramMiniC14.svg|center]]
|-----
! 15
| Z<sub>15</sub> = Z<sub>5</sub> × Z<sub>3</sub>
| Z<sub>5</sub>, Z<sub>3</sub> || multiplication of [[nimber]]s 1,...,15
| [[Image:GroupDiagramMiniC15.svg|center]]
|-----
! rowspan="5" | 16
| Z<sub>16</sub>
| Z<sub>8</sub>, Z<sub>4</sub>, Z<sub>2</sub>
| &nbsp;
| [[Image:GroupDiagramMiniC16.svg|center]]
|-----
| {{SubSup|Z|2|4}}
| Z<sub>2</sub> (15), {{SubSup|Z|2|2}} (35), {{SubSup|Z|2|3}} (15)</td>
| addition of nimbers 0,...,15
| [[Image:GroupDiagramMiniC2x4.svg|center]]
|-----
| Z<sub>4</sub> × {{SubSup|Z|2|2}}
| Z<sub>2</sub> (7), Z<sub>4</sub> (4), {{SubSup|Z|2|2}} (7), {{SubSup|Z|2|3}}, {{nowrap|Z<sub>4</sub> × Z<sub>2</sub>}} (6)
| &nbsp;
| [[Image:GroupDiagramMiniC2x2C4.svg|center]]
|-----
| Z<sub>8</sub> × Z<sub>2</sub>
| Z<sub>2</sub> (3), Z<sub>4</sub> (2), {{SubSup|Z|2|2}}, Z<sub>8</sub> (2), {{nowrap|Z<sub>4</sub> × Z<sub>2</sub>}}
| &nbsp;
| [[Image:GroupDiagramMiniC2C8.svg|center]]
|-----
| {{SubSup|Z|4|2}}
| Z<sub>2</sub> (3), Z<sub>4</sub> (6), {{SubSup|Z|2|2}}, {{nowrap|Z<sub>4</sub> × Z<sub>2</sub>}} (3)</td>
| &nbsp;
| [[Image:GroupDiagramMiniC4x2.svg|center]]
|}
 
== List of small non-abelian groups==
{| class="wikitable"
! Order
! Group
! Subgroups
! Properties
! [[Cycle graph (algebra)|Cycle graph]]
|-
! 6
| [[Dihedral group of order 6|S<sub>3</sub> = Dih<sub>3</sub>]]
| Z<sub>3</sub>, Z<sub>2</sub> (3)
| the smallest non-abelian group
| [[Image:GroupDiagramMiniD6.svg|center]]
|-
!rowspan="2"| 8
|
[[dihedral group of order 8|Dih<sub>4</sub>]]
| Z<sub>4</sub>, Z<sub>2</sub><sup>2</sup> (2), Z<sub>2</sub> (5)
|
| [[Image:GroupDiagramMiniD8.svg|center]]
|-
| [[quaternion group]], {{nowrap|1=Q<sub>8</sub> = Dic<sub>2</sub>}}
| Z<sub>4</sub> (3), Z<sub>2</sub>
| the smallest [[Hamiltonian group]]; smallest group demonstrating that all subgroups may be [[Normal subgroup|normal]] without the group being abelian; the smallest group ''G'' demonstrating that for a normal subgroup ''H'' the [[quotient group]] ''G''/''H'' need not be isomorphic to a subgroup of ''G''
| [[Image:GroupDiagramMiniQ8.svg|center]]
|-
! 10
| Dih<sub>5</sub>
| Z<sub>5</sub>, Z<sub>2</sub> (5)
|
| [[Image:GroupDiagramMiniD10.svg|center]]
|-
!rowspan="3"| 12
| Dih<sub>6</sub> = Dih<sub>3</sub> × Z<sub>2</sub>
| Z<sub>6</sub>, Dih<sub>3</sub> (2), Z<sub>2</sub><sup>2</sup> (3), Z<sub>3</sub>, Z<sub>2</sub> (7)
|
| [[Image:GroupDiagramMiniD12.svg|center]]
|-
| A<sub>4</sub>
| Z<sub>2</sub><sup>2</sup>, Z<sub>3</sub> (4), Z<sub>2</sub> (3)
| smallest group demonstrating that a group need not have a subgroup of every order that divides the group's order: no subgroup of order 6 (See [[Lagrange's theorem (group theory)|Lagrange's theorem]] and the [[Sylow theorems]].)
| [[Image:GroupDiagramMiniA4.svg|center]]
|-
| Dic<sub>3</sub> = Z<sub>3</sub> <!--&#x22CA;--> <math>\rtimes</math> Z<sub>4</sub>
| Z<sub>2</sub>, Z<sub>3</sub>, Z<sub>4</sub> (3), Z<sub>6</sub>
|
| [[Image:GroupDiagramMiniX12.svg|center]]
|-
! 14
| Dih<sub>7</sub>
| Z<sub>7</sub>, Z<sub>2</sub> (7)
|
| [[Image:GroupDiagramMiniD14.svg|center]]
|-
!rowspan="9"| 16<ref>Wild, Marcel. "[http://math.sun.ac.za/~wild/Marcel%20Wild%20-%20Home%20Page_files/Groups16AMM.pdf The Groups of Order Sixteen Made Easy]", [[American Mathematical Monthly]], Jan 2005</ref>
| Dih<sub>8</sub>
| Z<sub>8</sub>, Dih<sub>4</sub> (2), Z<sub>2</sub><sup>2</sup> (4), Z<sub>4</sub>, Z<sub>2</sub> (9)
|
| [[Image:GroupDiagramMiniD16.svg|center]]
|-
| Dih<sub>4</sub> × Z<sub>2</sub>
| Dih<sub>4</sub> (2), {{nowrap|Z<sub>4</sub> × Z<sub>2</sub>}}, Z<sub>2</sub><sup>3</sup> (2), Z<sub>2</sub><sup>2</sup> (11), Z<sub>4</sub> (2), Z<sub>2</sub> (11)
|
| [[Image:GroupDiagramMiniC2D8.svg|center]]
|-
| [[generalized quaternion group]], {{nowrap|1=Q<sub>16</sub> = Dic<sub>4</sub>}}
|
|
| [[Image:GroupDiagramMiniQ16.svg|center]]
|-
| Q<sub>8</sub> × Z<sub>2</sub>
|
| [[Hamiltonian group|Hamiltonian]]
| [[Image:GroupDiagramMiniC2Q8.svg|center]]
|-
| The order 16 [[quasidihedral group]]
|
|
| [[Image:GroupDiagramMiniQH16.svg|center]]
|-
| The order 16 [[Iwasawa group|modular group]]
|
|
| [[Image:GroupDiagramMiniC2C8.svg|center]]
|-
| Z<sub>4</sub> <!--&#x22CA;--> <math>\rtimes</math> Z<sub>4</sub>
|
|
| [[Image:GroupDiagramMinix3.svg|center]]
|-
| The group generated by the [[Pauli matrix|Pauli matrices]]
|
|
| [[Image:GroupDiagramMiniC2x2C4.svg|center]]
|-
| ''G''<sub>4,4</sub> = Z<sub>2</sub><sup>2</sup> <!--&#x22CA;--> <math>\rtimes</math> Z<sub>4</sub>
|
|
| [[Image:GroupDiagramMiniG44.svg|center]]
|}
 
==Small groups library==
The group theoretical [[computer algebra system]] [[GAP computer algebra system|GAP]] contains the "Small Groups library" which provides access to descriptions of small order groups. The groups are listed [[up to]] [[group isomorphism|isomorphism]]. At present, the library contains the following groups:<ref>Hans Ulrich Besche [http://www.icm.tu-bs.de/ag_algebra/software/small/ The Small Groups library]</ref>
* those of order at most 2000, except for order 1024 ({{gaps|423|164|062}} groups in the library; the ones of order 1024 had to be skipped, as there are an additional  {{gaps|49|487|365|422}} nonisomorphic [[p-group|2-groups]] of order 1024.);
* those of cubefree order at most 50000 (395 703 groups);
* those of squarefree order;
* those of order <math>p^n</math> for ''n'' at most 6 and ''p'' prime;
* those of order <math>p^7</math> for ''p'' = 3,5,7,11 (907 489 groups);
* those of order {{nowrap|''q''<sup>''n''</sup> × ''p''}} where ''q''<sup>''n''</sup> divides 2<sup>8</sup>, 3<sup>6</sup>, 5<sup>5</sup> or 7<sup>4</sup> and ''p'' is an arbitrary prime which differs from ''q'';
* those whose orders factorise into at most 3 primes.
It contains explicit descriptions of the available groups in computer readable format.
 
==See also==
*[[Classification of finite simple groups]], the four categories
*[[Composition series]], some groups may be broken down into simpler groups
*[[List of finite simple groups]]
*[[Finite_group#Number_of_groups_of_a_given_order|Number of groups of a given order]]
*[[Small Latin squares and quasigroups]]
 
== References ==
<references/>
 
==External links==
<!-- contents discontinued
*[http://www.math.usf.edu/~eclark/algctlg/small_groups.html Small groups]
-->
* [http://groupprops.subwiki.org/wiki/Category:Particular_groups Particular groups] groupprops.subwiki.org
* {{cite web
|first1= H. U.
|last1=Besche
|last2=Eick
|first2=B.
|last3=O'Brien
|first3=E.
|url=http://www.icm.tu-bs.de/ag_algebra/software/small/
|title=small group library
}}
* {{Cite document | first1=Marshall | last1=Hall, Jr. | authorlink1 = Marshall Hall (mathematician) | first2=James K. | last2=Senior | title=The Groups of Order 2<sup>''n''</sup> ({{nowrap|''n'' ≤ 6}}) | publisher=Macmillan | year=1964 | mr=168631  | postscript =. An exhaustive catalog of the 340 groups of order dividing 64 with detailed tables of defining relations, constants, and [[Lattice of subgroups|lattice]] presentations of each group in the notation the text defines. "Of enduring value to those interested in [[finite groups]]" (from the preface). | lccn=6416861}}
 
[[Category:Mathematics-related lists|Groups that are small]]
[[Category:Mathematical tables|Groups that are small]]
[[Category:Finite groups]]
 
{{Link FL|de}}

Latest revision as of 13:33, 29 December 2014

Journalists and Other Writers Elliott Janner from Marathon, has several hobbies and interests that include embroidery, health and fitness and church/church activities. Finds the charm in visiting destinations all over the planet, recently just coming back from Urnes Stave Church.

Feel free to visit my blog post :: Http://Smartchoicecart.In

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