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CyclicGroup[10]
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CyclicGroup[10]
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GroupMultiplicationTable[CyclicGroup[10]]
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MatrixPlot[%]
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GroupElements[CyclicGroup[8]]
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GroupGenerators[CyclicGroup[8]]
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{Cycles[{{1,2,3,4,5,6,7,8}}]}
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CayleyGraph[CyclicGroup[8]]
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PermutationList[#,8]&/@GroupElements[CyclicGroup[8]]
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{{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}}
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PermutationList[#,12]&/@GroupElements[CyclicGroup[8]]
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CyclicGroup[3]
{{1,2,3},{1,3,2},{2,1,3},{2,3,1}}
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FiniteGroupData[3]
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{{CyclicGroup,3}}
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FiniteGroupData[4]
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{{CyclicGroup,4},{AbelianGroup,{2,2}}}
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FiniteGroupData[5]
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{{CyclicGroup,5}}
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FiniteGroupData[6]
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{{AbelianGroup,{3,2}},{SymmetricGroup,3}}
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FiniteGroupData[7]
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{{CyclicGroup,7}}
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FiniteGroupData[8]
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{{CyclicGroup,8},{AbelianGroup,{4,2}},{AbelianGroup,{2,2,2}},{DihedralGroup,4},Quaternion}
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FiniteGroupData[12]
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{{AbelianGroup,{4,3}},{AbelianGroup,{3,2,2}},{AlternatingGroup,4},{DihedralGroup,6},{DicyclicGroup,3}}
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PermutationList[#,12]&/@GroupElements[AlternatingGroup[4]]
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PermutationList[#,4]&/@GroupElements[AlternatingGroup[4]]
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{{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}}]]
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{Cycles[{}],Cycles[{{2,3}}],Cycles[{{1,2}}],Cycles[{{1,2,3}}],Cycles[{{1,3,2}}],Cycles[{{1,3}}]}
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PermutationList[#,3]&/@%
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{{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
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GroupGenerators[SymmetricGroup[7]]
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{Cycles[{{1,2}}],Cycles[{{1,2,3,4,5,6,7}}]}
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GroupElementQ[PermutationGroup[{{2,3,1}}],{3,1,2}]
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True
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GroupElementQ[PermutationGroup[{{2,3,1}}],{2,1,3}]
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False
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??