Tetrahedron[]
In[]:=
Tetrahedron[]
Out[]=
Graphics3D[TruncatedPolyhedron[Tetrahedron[],1/3]]
In[]:=
Out[]=
Graphics3D/@NestList[TruncatedPolyhedron[#,1/2]&,Tetrahedron[],4]
In[]:=
Out[]=
Graphics3D/@NestList[TruncatedPolyhedron[#,1/3]&,Tetrahedron[],4]
In[]:=
Out[]=
GraphAutomorphismGroup[TorusGraph[{5,5}]]
In[]:=
Out[]=
GraphAutomorphismGroup

In[]:=
Out[]=
CayleyGraph
CayleyGraph[PermutationGroup[{Cycles[{{1,5,4}}],Cycles[{{3,4}}]}],VertexLabelsPlaced["Name",Center],VertexSize0.4]
In[]:=
Out[]=
GraphAutomorphismGroup[%]
In[]:=
Out[]=
CayleyGraph[SymmetricGroup[4]]
In[]:=
Out[]=
CayleyGraph[SymmetricGroup[5]]
In[]:=
Out[]=
CayleyGraph[SymmetricGroup[5]]
In[]:=
Out[]=
GroupGenerators[SymmetricGroup[6]]
In[]:=
{Cycles[{{1,2}}],Cycles[{{1,2,3,4,5,6}}]}
Out[]=
CayleyGraph[SymmetricGroup[6]]
In[]:=
Out[]=
gg=
;
In[]:=
NeighborhoodGraph[gg,1,2,GraphLayout"SpringElectricalEmbedding",VertexCoordinatesAutomatic]
In[]:=
Out[]=
NeighborhoodGraph[gg,100,2,GraphLayout"SpringElectricalEmbedding",VertexCoordinatesAutomatic]
In[]:=
Out[]=
Graph[EdgeList[NeighborhoodGraph[gg,100,2]]]
In[]:=
Out[]=
Graph[EdgeList[NeighborhoodGraph[gg,100,3]]]
In[]:=
Out[]=
Graph[EdgeList[NeighborhoodGraph[gg,100,5]]]
In[]:=
Out[]=
Graph[EdgeList[NeighborhoodGraph[gg,1,5]]]
In[]:=
Out[]=
GraphNeighborhoodVolumes[UndirectedGraph[gg],All,Automatic]
In[]:=
Out[]=
GraphNeighborhoodVolumes[UndirectedGraph[gg],{1},Automatic]
In[]:=
1{1,4,10,21,41,76,131,212,321,449,575,670,710,716,719,720}
Out[]=
Values[%]
In[]:=
{{1,4,10,21,41,76,131,212,321,449,575,670,710,716,719,720}}
Out[]=
First[%]
In[]:=
{1,4,10,21,41,76,131,212,321,449,575,670,710,716,719,720}
Out[]=
ListLogPlot[%433]
In[]:=
Out[]=
LogDifferences[%433]
In[]:=
Out[]=
N[%]
In[]:=
{2.,2.25985,2.57902,2.99829,3.38502,3.53202,3.60506,3.5222,3.18508,2.59518,1.75733,0.724454,0.113553,0.0606032,0.0215353}
Out[]=
FindSequenceFunction[%,n]
In[]:=
$Aborted
Out[]=
Ratios[%%]//N
In[]:=
{4.,2.5,2.1,1.95238,1.85366,1.72368,1.61832,1.51415,1.39875,1.28062,1.16522,1.0597,1.00845,1.00419,1.00139}
Out[]=
CayleyGraph[SymmetricGroup[6]]
Ca
Graph3D[%]
In[]:=
Out[]=
Table[Graph[EdgeList[CayleyGraph[SymmetricGroup[n]]]],{n,2,6}]
In[]:=
Out[]=
Table[Graph[EdgeList[CayleyGraph[AlternatingGroup[n]]],ImageSizeTiny],{n,3,7}]
In[]:=
Out[]=
TableGraphEdgeListNeighborhoodGraph
,1,n,ImageSizeTiny,{n,1,5}
In[]:=
Out[]=
AlternatingGroup
Table[Graph[EdgeList[CayleyGraph[CyclicGroup[n]]],ImageSizeTiny],{n,2,7}]
In[]:=
Out[]=
Table[Graph[EdgeList[CayleyGraph[DihedralGroup[n]]],ImageSizeTiny],{n,2,7}]
In[]:=
Out[]=
Direct product of cyclic groups:
Table[Graph[EdgeList[CayleyGraph[AbelianGroup[{n,n}]]],ImageSizeTiny],{n,2,7}]
In[]:=
Out[]=
Graph[EdgeList[CayleyGraph[MathieuGroupM11[]]]]
In[]:=
Out[]=
GraphPlot[%]
In[]:=
Out[]=
Table[Graph[EdgeList[NeighborhoodGraph[%458,1,n]],ImageSizeTiny],{n,1,8}]
In[]:=
Out[]=
GraphNeighborhoodVolumes[UndirectedGraph[CayleyGraph[SymmetricGroup[10]]],{1},Automatic]
In[]:=
$Aborted
Out[]=
GraphNeighborhoodVolumes[UndirectedGraph[CayleyGraph[SymmetricGroup[8]]],{1},Automatic]
In[]:=
1{1,4,10,22,45,89,169,311,558,969,1631,2650,4131,6190,8935,12400,16526,21159,26072,30849,35012,38091,39703,40191,40285,40310,40316,40319,40320}
Out[]=
Ratios[First[Values[%464]]]//N
In[]:=
{4.,2.5,2.2,2.04545,1.97778,1.89888,1.84024,1.79421,1.73656,1.68318,1.62477,1.55887,1.49843,1.44346,1.3878,1.33274,1.28035,1.23219,1.18322,1.13495,1.08794,1.04232,1.01229,1.00234,1.00062,1.00015,1.00007,1.00002}
Out[]=
ListLinePlot[%]
In[]:=
Out[]=
GraphNeighborhoodVolumes[UndirectedGraph[CayleyGraph[MathieuGroupM11[]]],{1},Automatic]
In[]:=
1{1,4,9,17,30,51,85,140,229,370,573,861,1281,1859,2629,3651,4841,6086,7117,7681,7890,7920}
Out[]=
Graph
,VertexLabelsAutomatic
In[]:=
Out[]=

https://en.wikipedia.org/wiki/Heisenberg_group

https://en.wikipedia.org/wiki/Gromov%27s_theorem_on_groups_of_polynomial_growth

In particular, Gromov’s theorem and the Bass–Guivarch formula imply that the order of polynomial growth of a finitely generated group is always either an integer or infinity (excluding for example, fractional powers).