In[]:=
CyclicGroup[10]
Out[]=
CyclicGroup[10]
In[]:=
GroupMultiplicationTable[CyclicGroup[10]]
Out[]=
In[]:=
MatrixPlot[%]
Out[]=
In[]:=
GroupElements[CyclicGroup[8]]
Out[]=
In[]:=
GroupGenerators[CyclicGroup[8]]
Out[]=
{Cycles[{{1,2,3,4,5,6,7,8}}]}
In[]:=
CayleyGraph[CyclicGroup[8]]
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In[]:=
PermutationList[#,8]&/@GroupElements[CyclicGroup[8]]
Out[]=
{{1,2,3,4,5,6,7,8},{2,3,4,5,6,7,8,1},{3,4,5,6,7,8,1,2},{4,5,6,7,8,1,2,3},{5,6,7,8,1,2,3,4},{6,7,8,1,2,3,4,5},{7,8,1,2,3,4,5,6},{8,1,2,3,4,5,6,7}}
In[]:=
PermutationList[#,12]&/@GroupElements[CyclicGroup[8]]
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https://mathoverflow.net/questions/190705/approximating-lie-groups-by-finite-groups
https://mathoverflow.net/questions/190705/approximating-lie-groups-by-finite-groups
https://arxiv.org/pdf/1712.01052.pdf
https://arxiv.org/pdf/1712.01052.pdf
CyclicGroup[3]
{{1,2,3},{1,3,2},{2,1,3},{2,3,1}}
In[]:=
FiniteGroupData[3]
Out[]=
{{CyclicGroup,3}}
In[]:=
FiniteGroupData[4]
Out[]=
{{CyclicGroup,4},{AbelianGroup,{2,2}}}
In[]:=
FiniteGroupData[5]
Out[]=
{{CyclicGroup,5}}
In[]:=
FiniteGroupData[6]
Out[]=
{{AbelianGroup,{3,2}},{SymmetricGroup,3}}
In[]:=
FiniteGroupData[7]
Out[]=
{{CyclicGroup,7}}
In[]:=
FiniteGroupData[8]
Out[]=
{{CyclicGroup,8},{AbelianGroup,{4,2}},{AbelianGroup,{2,2,2}},{DihedralGroup,4},Quaternion}
In[]:=
FiniteGroupData[12]
Out[]=
{{AbelianGroup,{4,3}},{AbelianGroup,{3,2,2}},{AlternatingGroup,4},{DihedralGroup,6},{DicyclicGroup,3}}
In[]:=
PermutationList[#,12]&/@GroupElements[AlternatingGroup[4]]
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In[]:=
PermutationList[#,4]&/@GroupElements[AlternatingGroup[4]]
Out[]=
{{1,2,3,4},{1,3,4,2},{1,4,2,3},{2,1,4,3},{2,3,1,4},{2,4,3,1},{3,1,2,4},{3,2,4,1},{3,4,1,2},{4,1,3,2},{4,2,1,3},{4,3,2,1}}
Forming a group
Forming a group
In[]:=
GroupElements[PermutationGroup[{{1,2,3},{1,3,2},{2,1,3},{2,3,1}}]]
Out[]=
{Cycles[{}],Cycles[{{2,3}}],Cycles[{{1,2}}],Cycles[{{1,2,3}}],Cycles[{{1,3,2}}],Cycles[{{1,3}}]}
In[]:=
PermutationList[#,3]&/@%
Out[]=
{{1,2,3},{1,3,2},{2,1,3},{2,3,1},{3,1,2},{3,2,1}}
This is symmetric group on 3 elements....
Rank is minimum number of generators
Rank is minimum number of generators
In[]:=
GroupGenerators[SymmetricGroup[7]]
Out[]=
{Cycles[{{1,2}}],Cycles[{{1,2,3,4,5,6,7}}]}
In[]:=
GroupElementQ[PermutationGroup[{{2,3,1}}],{3,1,2}]
Out[]=
True
Random permutations
Random permutations
Representations of groups
Representations of groups
Permutation representation of a group vs. matrix representation vs. generators & relations
Permutation representation of a group vs. matrix representation vs. generators & relations
Matrix representations
Matrix representations
E.g. SO(3) :
Permutation representations
Permutation representations
The thing generated by:
Cyclic Case
Cyclic Case
Imagine that we took Lie group space and made a lattice out of it...
Imagine that we took Lie group space and made a lattice out of it...
Then simply for permutations of elements to elements
Any given rotation at least approximately maps these points to other points....
Pick a random rotation matrix....
Can we do GroupOrbits??
Can we do GroupOrbits??
What is the closest permutation
What should the norm of closest permutation be??