Finite Group

Author

Eric W. Weisstein
March 20, 2012

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GroupTheory/FiniteGroup.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FiniteGroup.html.

©2012 Wolfram Research, Inc. except for portions noted otherwise

Counts

Package

?NumberOfFiniteGroups

NumberOfFiniteGroups/@Range[2048]

Table

DisplayForm[GridBox[Flatten/@Transpose@Partition[Table[{n,NumberOfFiniteGroups[n],NumberOfAbelianGroups[n]},{n,250}],50],
GridFrame->True,RowLines->True,ColumnLines->True,ColumnAlignments->Right]]

Built-In

?FiniteGroupCount

FiniteGroupCount[Range[2000]]==NumberOfFiniteGroups/@Range[2000]

V7

FiniteGroupCount[Range[2000]]==NumberOfFiniteGroups/@Range[2000]

Position[FiniteGroupCount[Range[2500]],_FiniteGroupCount]//Flatten

V9

Select[Table[{n,Length[FiniteGroupData[n]],FiniteGroupCount[n]},{n,2050}],#[[2]]=!=#[[3]]&]//TableForm

Position[FiniteGroupCount[Range[3000]],_FiniteGroupCount]//Flatten

Generating Functions

Generating Function

Sum[FiniteGroupCount[n]x^n,{n,∞}]

GeneratingFunction[FiniteGroupCount[n],n,z]

FindGeneratingFunction[{1,1,1,2,1,2,1,5,2,2,1,5,1,2,1,14,1,5,1,5,2,2,1,15,2,2,5,4,1,4,1,51,1,2,1,14,1,2,2,14,1,6,1,4,2,2},x]

ExponentialGeneratingFunction

Sum[FiniteGroupCount[n]((x^n)/(n!)),{n,∞}]

DirichletGeneratingFunction

Sum[((FiniteGroupCount[n])/(n^s)),{n,∞}]

Lambert Series

Sum[((FiniteGroupCount[n](x^n))/(1-(x^n))),{n,∞}]

Numbers of Nonisomorphic Finite Groups of Order 1, 2, …

Timing[a=Table[NumberOfFiniteGroups[n],{n,2000}];]

Smallest Missing Number of Nonisomorphic Finite groups

max=Position[Equal@@@Transpose[{u=Union[a],Range[Length[u]]}],False,1,1][[1,1]]

Smallest Orders having 1, 2, 3, … Abelian groups

orders=Rest[First/@Split[FoldList[Max,0,a]]]

Flatten[Position[a,#,1,1]&/@orders]

Special Cases

Refine[FiniteGroupCount[n],n==1]

Refine[FiniteGroupCount[p],p∈Primes]

Table[FiniteGroupCount[Prime[n]],{n,100}]//Union

Refine[FiniteGroupCount[p q],(p|q)∈Primes&&q>p]

l=Sort/@Select[Partition[Prime[RandomInteger[{1,1000},{10000}]],2],Unequal@@#&];

And@@Equal@@@(Function[{p,q},{FiniteGroupCount[p q],If[Mod[q-1,p]==0,2,1]}]@@@l)

Refine[FiniteGroupCount[p^2],p∈Primes]

Table[FiniteGroupCount[Prime[n]^2],{n,100}]//Union

Refine[FiniteGroupCount[p^3],p∈Primes]

Table[FiniteGroupCount[Prime[n]^3],{n,100}]//Union

Generator Permutations for Groups

sg ={{ (* order 2*)
{{2,1}}
},{ (* order 3*)
{{2,3,1}}
},{ (* order 4*)
{{3,4,2,1},{2,1,4,3}},
{{3,4,1,2},{2,1,4,3}}
},{ (* order 5*)
{{2,3,4,5,1}}
},{ (* order 6*)
{{4,6,5,1,3,2},{2,3,1,5,6,4}},{{4,5,6,1,2,3},{2,3,1,5,6,4}}
},{ (* order 7*)
{{2,3,4,5,6,7,1}}
},{ (* order 8*)
{{5,6,7,8,3,4,2,1},{3,4,2,1,7,8,6,5},{2,1,4,3,6,5,8,7}},{{5,6,7,8,2,1,4,3},{3,4,1,2,7,8,5,6},{2,1,4,3,6,5,8,7}},{{5,6,8,7,1,2,4,3},{3,4,1,2,7,8,5,6},{2,1,4,3,6,5,8,7}},{{5,6,8,7,2,1,3,4},{3,4,2,1,7,8,6,5},{2,1,4,3,6,5,8,7}},{{5,6,7,8,1,2,3,4},{3,4,1,2,7,8,5,6},{2,1,4,3,6,5,8,7}}
},{ (* order 9*)
},{ (* order 10*)
{{6,10,9,8,7,1,5,4,3,2},{2,3,4,5,1,7,8,9,10,6}},{{6,7,8,9,10,1,2,3,4,5},{2,3,4,5,1,7,8,9,10,6}}
},{ (* order 11*)
{RotateLeft[Range[11]]}
},{ (* order 12*)
{{7,9,8,10,12,11,4,6,5,1,3,2},{4,5,6,1,2,3,10,11,12,7,8,9},{2,3,1,5,6,4,8,9,7,11,12,10}},{{7,8,9,10,11,12,2,1,4,3,6,5},{3,4,5,6,1,2,9,10,11,12,7,8},{2,1,4,3,6,5,8,7,10,9,12,11}},{{5,8,6,7,9,12,10,11,1,4,2,3},{3,4,1,2,7,8,5,6,11,12,9,10},{2,1,4,3,6,5,8,7,10,9,12,11}},{{7,9,8,10,12,11,1,3,2,4,6,5},{4,5,6,1,2,3,10,11,12,7,8,9},{2,3,1,5,6,4,8,9,7,11,12,10}},{{7,8,9,10,11,12,1,2,3,4,5,6},{4,5,6,1,2,3,10,11,12,7,8,9},{2,3,1,5,6,4,8,9,7,11,12,10}}
},{ (* order 13*)
{RotateLeft[Range[13]]}
},{ (* order 14*)
{{8,14,13,12,11,10,9,1,7,6,5,4,3,2},{2,3,4,5,6,7,1,9,10,11,12,13,14,8}},{{8,9,10,11,12,13,14,1,2,3,4,5,6,7},{2,3,4,5,6,7,1,9,10,11,12,13,14,8}}
},{ (* order 15*)
},{ (* order 16*)
{{9,10,11,12,13,14,15,16,5,6,7,8,3,4,2,1},{5,6,7,8,3,4,2,1,13,14,15,16,11,12,10,9},{3,4,2,1,7,8,6,5,11,12,10,9,15,16,14,13},{2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15}},{{9,10,11,12,13,14,15,16,3,4,1,2,7,8,5,6},{5,6,7,8,2,1,4,3,13,14,15,16,10,9,12,11},{3,4,1,2,7,8,5,6,11,12,9,10,15,16,13,14},{2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15}},{{9,10,11,12,15,16,13,14,2,1,4,3,8,7,6,5},{5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12},{3,4,1,2,7,8,5,6,11,12,9,10,15,16,13,14},{2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15}},{{9,10,11,12,15,16,13,14,2,1,4,3,8,7,6,5},{5,6,7,8,3,4,1,2,13,14,15,16,11,12,9,10},{3,4,1,2,7,8,5,6,11,12,9,10,15,16,13,14},{2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15}},{{9,10,11,12,13,14,15,16,3,4,2,1,7,8,6,5},{5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12},{3,4,2,1,7,8,6,5,11,12,10,9,15,16,14,13},{2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15}},{{9,10,11,12,14,13,16,15,3,4,2,1,8,7,5,6},{5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12},{3,4,2,1,7,8,6,5,11,12,10,9,15,16,14,13},{2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15}},{{9,10,12,11,15,16,13,14,1,2,4,3,7,8,5,6},{5,6,8,7,1,2,4,3,13,14,16,15,9,10,12,11},{3,4,2,1,7,8,6,5,11,12,10,9,15,16,14,13},{2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15}},{{9,10,12,11,15,16,13,14,2,1,3,4,8,7,6,5},{5,6,8,7,1,2,4,3,13,14,16,15,9,10,12,11},{3,4,2,1,7,8,6,5,11,12,10,9,15,16,14,13},{2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15}},{{9,10,12,11,15,16,13,14,2,1,3,4,8,7,6,5},{5,6,8,7,2,1,3,4,13,14,16,15,10,9,11,12},{3,4,2,1,7,8,6,5,11,12,10,9,15,16,14,13},{2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15}},{{9,10,11,12,13,14,15,16,2,1,4,3,6,5,8,7},{5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12},{3,4,1,2,7,8,5,6,11,12,9,10,15,16,13,14},{2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15}},{{9,10,11,12,14,13,16,15,1,2,3,4,6,5,8,7},{5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12},{3,4,1,2,7,8,5,6,11,12,9,10,15,16,13,14},{2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15}},{{9,10,11,12,14,13,16,15,2,1,4,3,5,6,7,8},{5,6,7,8,2,1,4,3,13,14,15,16,10,9,12,11},{3,4,1,2,7,8,5,6,11,12,9,10,15,16,13,14},{2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15}},{{9,10,11,12,14,13,16,15,1,2,3,4,6,5,8,7},{5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12},{3,4,2,1,7,8,6,5,11,12,10,9,15,16,14,13},{2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15}},{{9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8},{5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12},{3,4,1,2,7,8,5,6,11,12,9,10,15,16,13,14},{2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15}}
},{ (* order 17*)
{RotateLeft[Range[17]]}
},{ (* order 18*)
{{10,12,11,17,16,18,14,13,15,1,3,2,8,7,9,5,4,6},{4,5,6,7,8,9,3,1,2,13,14,15,16,17,18,12,10,11},{2,3,1,5,6,4,8,9,7,11,12,10,14,15,13,17,18,16}},{{10,11,12,13,14,15,16,17,18,1,2,3,4,5,6,7,8,9},{4,5,6,7,8,9,2,3,1,13,14,15,16,17,18,11,12,10},{2,3,1,5,6,4,8,9,7,11,12,10,14,15,13,17,18,16}},{{10,12,11,13,15,14,16,18,17,1,3,2,4,6,5,7,9,8},{4,5,6,7,8,9,1,2,3,13,14,15,16,17,18,10,11,12},{2,3,1,5,6,4,8,9,7,11,12,10,14,15,13,17,18,16}},{{10,12,11,16,18,17,13,15,14,1,3,2,7,9,8,4,6,5},{4,5,6,7,8,9,1,2,3,13,14,15,16,17,18,10,11,12},{2,3,1,5,6,4,8,9,7,11,12,10,14,15,13,17,18,16}},{{10,11,12,13,14,15,16,17,18,1,2,3,4,5,6,7,8,9},{4,5,6,7,8,9,1,2,3,13,14,15,16,17,18,10,11,12},{2,3,1,5,6,4,8,9,7,11,12,10,14,15,13,17,18,16}}
},{ (* order 19*)
{RotateLeft[Range[19]]}
},{ (* order 20*)
{{11,15,14,13,12,16,20,19,18,17,6,10,9,8,7,1,5,4,3,2},{6,7,8,9,10,1,2,3,4,5,16,17,18,19,20,11,12,13,14,15},{2,3,4,5,1,7,8,9,10,6,12,13,14,15,11,17,18,19,20,16}},{{11,12,13,14,15,16,17,18,19,20,2,1,4,3,6,5,8,7,10,9},{3,4,5,6,7,8,9,10,1,2,13,14,15,16,17,18,19,20,11,12},{2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15,18,17,20,19}},{{11,13,15,12,14,16,18,20,17,19,6,8,10,7,9,1,3,5,2,4},{6,10,9,8,7,1,5,4,3,2,16,20,19,18,17,11,15,14,13,12},{2,3,4,5,1,7,8,9,10,6,12,13,14,15,11,17,18,19,20,16}},{{11,15,14,13,12,16,20,19,18,17,1,5,4,3,2,6,10,9,8,7},{6,7,8,9,10,1,2,3,4,5,16,17,18,19,20,11,12,13,14,15},{2,3,4,5,1,7,8,9,10,6,12,13,14,15,11,17,18,19,20,16}},{{11,12,13,14,15,16,17,18,19,20,1,2,3,4,5,6,7,8,9,10},{6,7,8,9,10,1,2,3,4,5,16,17,18,19,20,11,12,13,14,15},{2,3,4,5,1,7,8,9,10,6,12,13,14,15,11,17,18,19,20,16}}
},{ (* order 21*)
},{ (* order 22*)
},{ (* order 23*)
},{ (* order 24*)
},{ (* order 25*)
},{ (* order 26*)
},{ (* order 27*)
},{ (* order 28*)
},{ (* order 29*)
},{ (* order 30*)
},{ (* order 31*)
},{ (* order 32*)
},{ (* order 33*)
},{ (* order 34*)
},{ (* order 35*)
},{ (* order 36*)
},{ (* order 37*)
},{ (* order 38*)
},{ (* order 39*)
},{ (* order 40*)
},{ (* order 41*)
},{ (* order 42*)
},{ (* order 44*)
},{ (* order 45*)
},{ (* order 46*)
},{ (* order 47*)
},{ (* order 48*)
},{ (* order 49*)
},{ (* order 50*)
},{ (* order 51*)
},{ (* order 52*)
}};

Length/@sg

Map[RecognizeGraph[CayleyGraph[#]]&,DeleteCases[sg,{}],{2}]//Column