Le texto fonte de Tabella de parve gruppos

La prime tabella lista le gruppos finite con un ordine minor o equal a 20 excepte isomorphitate.

{| class="wikitable sorttable" style="text-align:right"
! style="text-align:left"| Ordine 
||'''1'''||'''2'''||'''3'''||'''4'''||'''5'''||'''6'''||'''7'''||'''8'''||'''9'''||'''10'''||'''11'''||'''12'''||'''13'''||'''14'''||'''15'''||'''16'''||'''17'''||'''18'''||'''19'''||'''20'''
|-
! style="text-align:left"| Quantitate de gruppos abelian
|1||1||1||2||1||1||1||3||2||1||1||2||1||1||1||5||1||2||1||2
|-
! style="text-align:left"| Quantitate de gruppos non-abelian
|0||0||0||0||0||1||0||2||0||1||0||3||0||1||0||9||0||3||0||3
|-
! style="text-align:left"| Quantitate de gruppos in toto
|1||1||1||2||1||2||1||5||2||2||1||5||1||2||1||14||1||5||1||5
|}

== Abbreviaturas ==
* <math>A_2 \cong S_1\cong \Z_1</math> es le gruppo trivial.
* <math>A_n</math> es le [[gruppo alternante]] del grado <math>n</math>, con <math>n!/2</math> permutationes de <math>n</math> elementos pro <math>n\ge2</math>.
* <math>C_n</math> = <math>\Z_n</math> = <math>\Z / n\Z</math> es le [[gruppo cyclic]] del ordine <math>n</math>.
* <math>D_n</math> es le [[gruppo dihedre]] del ordine <math>2n</math>.
* <math>\mathrm{Dic}_n</math> es le [[gruppo dicyclic]] del ordine <math>4n</math>.
* <math>Q_8</math> es le [[gruppo de quaterniones]] del ordine <math>8</math>.
* <math>S_n</math> es le [[gruppo symmetric]] del grado <math>n</math>, con <math>n!</math> permutationes de <math>n</math> elementos.
* <math>V_4</math> es le [[gruppo de Klein]] del ordine <math>4</math>.

== Tabella ==
{| class="wikitable"
|-
! Ordine
! Gruppo
! Subgruppos non-trivial
! Proprietates
! Graphico cyclo
|-
! 1
| <math>\Z_1\cong S_1\cong A_2</math>
| 
| abelian, cyclic
| [[File:GroupDiagramMiniC1.svg|center]]
|-
! 2
| <math>\Z_2\cong S_2\cong D_1</math>
|
| abelian, finite, simple, cyclic, le minus grande gruppo non-trivial
| [[File:GroupDiagramMiniC2.svg|center]]
|-
! 3
| <math>\Z_3\cong A_3</math>
|
| abelian, simple, cyclic
| [[File:GroupDiagramMiniC3.svg|center]]
|-
! rowspan="2"| 4
| <math>\Z_4\cong \mathrm{Dic}_1</math>
| <math>\Z_2</math>
| abelian, cyclic
| [[File:GroupDiagramMiniC4.svg|center]]
|-
| <math>V_4\cong\Z_2^2\cong D_2</math>
| <math>3\cdot\Z_2</math>
| abelian, le minus grande gruppo non-cyclic
| [[File:GroupDiagramMiniD4.svg|center]]
|-
! 5
| <math>\Z_5</math>
|
| abelian, simple, cyclic
| [[File:GroupDiagramMiniC5.svg|center]]
|-
! rowspan="2"| 6
| <math>\Z_6\cong\Z_2\times\Z_3</math>
| <math>\Z_3</math>, <math>\Z_2</math>
| abelian, cyclic
| [[File:GroupDiagramMiniC6.svg|center]]
|-
| <math>S_3\cong D_3</math> &nbsp; gruppo symetric <math>S_3</math>
| <math>\Z_3</math>, <math>3 \cdot \Z_2</math>
| le minus grande gruppo non-ablian
| [[File:GroupDiagramMiniD6.svg|center]]
|-
! 7
| <math>\Z_7</math>
|
| abelian, simple, cyclic
| [[File:GroupDiagramMiniC7.svg|center]]
|-
! rowspan="5"| 8
| <math>\Z_8</math>
| <math>\Z_4</math>, <math>\Z_2</math>
| abelian, cyclic
| [[File:GroupDiagramMiniC8.svg|center]]
|-
| <math>\Z_2\times\Z_4</math>
| <math>2\cdot\Z_4</math>, <math>3\cdot\Z_2</math>, <math>D_2</math>
| abelian
| [[File:GroupDiagramMiniC2C4.svg|center]]
|-
| <math>\Z_2^3\cong D_2\times\Z_2</math>
| <math>7 \cdot \Z_2</math>, <math>7\cdot D_2</math>
| abelian
| [[File:GroupDiagramMiniC2x3.svg|center]]
|-
| <math>D_4</math>
| <math>\Z_4</math>, <math>2\cdot D_2</math>, <math>5\cdot\Z_2</math>
| non-abelian
| [[File:GroupDiagramMiniD8.svg|center]]
|-
| <math>Q_8\cong \mathrm{Dic}_2</math>
| <math>3\cdot\Z_4</math>, <math>\Z_2</math>
| non-abelian; le minus grande [[gruppo hamiltonian]]
| [[File:GroupDiagramMiniQ8.svg|center]]
|-
! rowspan="2"| 9
| <math>\Z_9</math>
| <math>\Z_3</math>
| abelian, cyclic
| [[File:GroupDiagramMiniC9.svg|center]]
|-
| <math>\Z_3^2</math>
| <math>4\cdot\Z_3</math>
| abelian
| [[File:GroupDiagramMiniC3x2.svg|center]]
|-
! rowspan="2"| 10
| <math>\Z_{10}\cong\Z_2\times\Z_5</math>
| <math>\Z_5</math>, <math> \Z_2</math>
| abelian, cyclic
| [[File:GroupDiagramMiniC10.svg|center]]
|-
| <math>D_5</math>
| <math>\Z_5</math>, <math>5\cdot\Z_2</math>
| non-abelian
| [[File:GroupDiagramMiniD10.svg|center]]
|-
! 11
| <math>\Z_{11}</math>
|
| abelian, simple, cyclic
| [[File:GroupDiagramMiniC11.svg|center]]
|-
! rowspan="5"| 12
| <math>\Z_{12}\cong\Z_4\times\Z_3</math>
| <math>\Z_6</math>, <math>\Z_4</math>, <math>\Z_3</math>, <math>\Z_2</math>
| abelian, cyclic
| [[File:GroupDiagramMiniC12.svg|center]]
|-
<!--{{groups 12na}}-->
| <math>\Z_2\times\Z_6\cong\Z_2^2\times\Z_3\cong D_2\times\Z_3</math>
| <math>3 \cdot \Z_6</math>, <math>\Z_3</math>, <math>D_2</math>, <math>3 \cdot \Z_2</math>
| abelian
| [[File:GroupDiagramMiniC2C6.svg|center]]
|-
| <math>D_6\cong D_3\times\Z_2</math>
| <math>\Z_6</math>, <math>2 \cdot D_3</math>, <math>3 \cdot D_2</math>, <math>\Z_3</math>, <math>7 \cdot \Z_2</math>
| non-abelian
| [[File:GroupDiagramMiniD12.svg|center]]
|-
| <math>A_4</math>
| <math>D_2</math>, <math>4\cdot\Z_3</math>, <math>3\cdot\Z_2</math>
| non-abelian; nulle subgruppo de ordine 6
| [[File:GroupDiagramMiniA4.svg|center]]
|-
| <math>\mathrm{Dic}_3 </math>
| <math>\Z_6</math>, <math>3\cdot\Z_4</math>, <math>\Z_3</math>, <math>\Z_2</math>
| non-abelian
| [[File:GroupDiagramMiniX12.svg|center]]
|-
! 13
| <math>\Z_{13}</math>
|
| abelian, simple, cyclic
| [[File:GroupDiagramMiniC13.svg|center]]
|-
! rowspan="2"| 14
| <math>\Z_{14}\cong\Z_2\times\Z_7</math>
| <math>\Z_7</math>, <math> \Z_2</math>
| abelian, cyclic
| [[File:GroupDiagramMiniC14.svg|center]]
|-
| <math>D_7</math>
| <math>\Z_7</math>, <math> 7\cdot\Z_2</math>
| non-abelian
| [[File:GroupDiagramMiniD14.svg|center]]
|-
! 15
| <math>\Z_{15}\cong\Z_3\times\Z_5</math>
| <math>\Z_5</math>, <math> \Z_3</math>
| abelian, cyclic
| [[File:GroupDiagramMiniC15.svg|center]]
|-
! rowspan="14"| 16
<!--{{groups 16a|abelian<br><br>}}-->
| <math>\Z_{16}</math>
| <math>\Z_8</math>, <math>\Z_4</math>, <math>\Z_2</math>
| abelian, cyclic
| [[File:GroupDiagramMiniC16.svg|center]]
|-
| <math>\Z_2^4</math>
| <math>15 \cdot \Z_2</math>, <math>35 \cdot D_2</math>, <math>15 \cdot \Z_2^3</math>
| abelian
| [[File:GroupDiagramMiniC2x4.svg|center]]
|-
| <math>\Z_4\times\Z_2^2</math>
| <math>7 \cdot \Z_2</math>, <math>4 \cdot \Z_4</math>, <math>7 \cdot D_2</math>, <math>\Z_2^3</math>, <math>6 \cdot \Z_4 \times \Z_2</math>
| abelian
| [[File:GroupDiagramMiniC2x2C4.svg|center]]
|-
| <math>\Z_8\times\Z_2</math>
| <math>3 \cdot \Z_2</math>, <math>2 \cdot \Z_4</math>, <math>D_2</math>, <math>2 \cdot \Z_8</math>, <math>\Z_4 \times \Z_2</math>
| abelian
| [[File:GroupDiagramMiniC2C8.svg|center]]
|-
| <math>\Z_4^2</math>
| <math>3 \cdot \Z_2</math>, <math>6 \cdot \Z_4</math>, <math> D_2</math>,<math> 3 \cdot \Z_4 \times \Z_2</math>
| abelian
| [[File:GroupDiagramMiniC4x2.svg|center]]
|-
| <math>D_8</math>
| <math>\Z_8</math>, <math>2 \cdot D_4</math>, <math>4 \cdot D_2</math>, <math>\Z_4</math>, <math>9 \cdot \Z_2</math>
| non-abelian
| [[File:GroupDiagramMiniD16.svg|center]]
|-
| <math>D_4\times\Z_2</math>
| <math>4 \cdot D_4</math>, <math>\Z_4 \times \Z_2</math>, <math>2 \cdot \Z_2^3</math>, <math>13 \cdot \Z_2^2</math>, <math>2 \cdot \Z_4</math>, <math>11 \cdot \Z_2</math>
| non-abelian
| [[File:GroupDiagramMiniC2D8.svg|center]]
|-
| <math>Q_{16}\cong \mathrm{Dic_4}</math>
| <math>\Z_8</math>, <math>2 \cdot Q_8</math>, <math>5 \cdot \Z_4</math>, <math>\Z_2</math>
| non-abelian
| [[File:GroupDiagramMiniQ16.svg|center]]
|-
| <math>Q_8\times\Z_2</math>
| <math>3 \cdot \Z_2 \times \Z_4</math>, <math>4 \cdot Q_8</math>, <math>6 \cdot \Z_4</math>, <math>\Z_2 \times \Z_2</math>, <math>3 \cdot \Z_2</math>
| non-abelian, [[gruppo hamiltonian]]
| [[File:GroupDiagramMiniC2Q8.svg|center]]
|-
| gruppo quasi-dihedre
| <math>\Z_8</math>, <math>Q_8</math>, <math>D_4</math>, <math>3\cdot\Z_4</math>, <math>2\cdot\Z_2\times\Z_2</math>, <math>5\cdot\Z_2</math>
| non-abelian
| [[File:GroupDiagramMiniQH16.svg|center]]
|-
| M-gruppo (gruppo non-abelian, non-hamiltonian, modular)
| <math>2\cdot\Z_8</math>, <math>\Z_4\times\Z_2</math>, <math>2\cdot\Z_4</math>, <math>\Z_2\times\Z_2</math>, <math>3\cdot\Z_2</math>
| non-abelian
| [[File:GroupDiagramMiniC2C8.svg|center]]
|-
| producto semidirecte <math>\Z_4 \rtimes \Z_4</math>
| <math>3\cdot\Z_2\times\Z_4</math>, <math>6\cdot\Z_4</math>, <math>\Z_2\times\Z_2</math>, <math>3\cdot\Z_2</math>
| non-abelian
| [[File:GroupDiagramMinix3.svg|center]]
|-
| le gruppo create per matrices de Pauli
| <math>3\cdot\Z_2\times\Z_4</math>, <math>3\cdot D_4</math>, <math>Q_8</math>, <math>4\cdot \Z_4</math>, <math>3\cdot\Z_2\times\Z_2</math>, <math>7\cdot\Z_2</math>
| non-abelian
| [[File:GroupDiagramMiniC2x2C4.svg|center]]
|-
| <math>G_{4,4} = V_4 \rtimes \Z_4</math>
| <math>2\cdot\Z_2\times\Z_4</math>, <math>\Z_2\times\Z_2\times\Z_2</math>, <math>4\cdot\Z_4</math>, <math>7\cdot\Z_2\times\Z_2</math>, <math>7\cdot\Z_2</math>
| non-abelian
| [[File:GroupDiagramMiniG44.svg|center]]
|-
! 17
| <math>\Z_{17}</math>
|
| abelian, simple, cyclic
| [[File:GroupDiagramMiniC17.svg|70px|center]]
|-
! rowspan="5"| 18
| <math>\Z_{18} \cong \Z_9\times \Z_2</math>
| <math>\Z_9, \Z_6, \Z_3, \Z_2</math>
| abelian, cyclic
| [[File:GroupDiagramMiniC18.svg|70px|center]]
|-
| <math>\Z_6 \times \Z_3</math>
| <math>\Z_6, \Z_3, \Z_2</math>
| abelian
| [[File:GroupDiagramMiniC3C6.png|70px|center]]
|-
| <math>D_9</math>
|
| non-abelian
| [[File:GroupDiagramMiniD18.png|70px|center]]
|-
| <math>S_3 \times \Z_3</math>
|
| non-abelian
| [[File:GroupDiagramMiniC3D6.png|70px|center]]
|-
| <math>(\Z_3 \times \Z_3)\rtimes_\alpha \Z_2</math> con <math>\alpha(1)=\begin{pmatrix}2&0\\0&2\end{pmatrix}</math>
|
| non-abelian
| [[File:GroupDiagramMiniG18-4.png|70px|center]]
|-
! 19
| <math>\Z_{19}</math>
|
| abelian, simple, cyclic
| [[File:GroupDiagramMiniC19.svg|70px|center]]
|-
! rowspan="5"| 20
| <math>\Z_{20} \cong \Z_5 \times \Z_4</math>
| <math>\Z_{10}, \Z_5, \Z_4, \Z_2</math>
| abelian, cyclic
| [[File:GroupDiagramMiniC20.svg|70px|center]]
|-
| <math>\Z_{10}\times \Z_2 \cong \Z_5\times \Z_2 \times \Z_2</math>
| <math>\Z_5, \Z_2</math>
| abelian
| [[File:GroupDiagramMiniC2C10.png|70px|center]]
|-
| <math>Q_{20} \cong \mathrm{Dic}_5</math>
|
| non-abelian
| [[File:GroupDiagramMiniQ20.png|100px|center]]
|-
| <math>\Z_5 \rtimes \Z_4 \cong</math> gruppo affine <math>\mathrm{AGL}_1</math>(5)
|
| non-abelian
| [[File:GroupDiagramMiniC5semiprodC4.png|70px|center]]
|-
| <math>D_{10} \cong D_5\times \Z_2</math>
| <math>\Z_{10}, D_5, \Z_5, 5\cdot V_4, 6\cdot \Z_2</math>
| non-abelian
| [[File:GroupDiagramMiniD20.png|70px|center]]
|}

[[Categoria:Theoria de gruppos]]
[[Categoria:Listas mathematic]]