ApCoCoA-1:Alternating groups
From ApCoCoAWiki
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
Description
The alternating groups is the group of all even permutations of a finite set. Every alternating group is a subgroups of the correspondent symmetric group. A finite representation is given by:
A_{n+2} = <x_{1},..x_{n} | x_{i}^{3} = (x_{i}x_{j})^2 = 1 for every i != j>
Reference
PRESENTATIONS OF FINITE SIMPLE GROUPS: A COMPUTATIONAL APPROACH R. M. GURALNICK, W. M. KANTOR, M. KASSABOV, AND A. LUBOTZKY
Computation
/*Use the ApCoCoA package ncpoly.*/ // Number of alternating group MEMORY.N:=3; Use ZZ/(2)[a[1..MEMORY.N]]; NC.SetOrdering("LLEX"); Define CreateRelationsAlternating() Relations:=[]; // add the relation a_{i}^{3} = 1 For Index0 := 1 To MEMORY.N Do Append(Relations,[[a[Index0]^3],[1]]); EndFor; // add the relation (a_{i}a_{j})^2 = 1 for every i != j For Index1 := 1 To MEMORY.N Do For Index2 := 1 To MEMORY.N Do If (Index1 <> Index2) Then Append(Relations,[[a[Index1],a[Index2],a[Index1],a[Index2]],[1]]); EndIf; EndFor; EndFor; Return Relations; EndDefine; Relations:=CreateRelationsAlternating(); Gb:=NC.GB(Relations);
Examples in Symbolic Data Format
Alternating group 3
<FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier"> <vars>a1,a2,a3</vars> <basis> <ncpoly>a1^3-1</ncpoly> <ncpoly>a2^3-1</ncpoly> <ncpoly>a3^3-1</ncpoly> <ncpoly>a1*a2*a1*a2-1</ncpoly> <ncpoly>a1*a3*a1*a3-1</ncpoly> <ncpoly>a2*a1*a2*a1-1</ncpoly> <ncpoly>a2*a3*a2*a3-1</ncpoly> <ncpoly>a3*a1*a3*a1-1</ncpoly> <ncpoly>a3*a2*a3*a2-1</ncpoly> </basis> <Comment>Alternating_group_3</Comment> </FREEALGEBRA>
Alternating group 4
<FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier"> <vars>a1,a2,a3,a4</vars> <uptoDeg>6</uptoDeg> <basis> <ncpoly>a1^3-1</ncpoly> <ncpoly>a2^3-1</ncpoly> <ncpoly>a3^3-1</ncpoly> <ncpoly>a4^3-1</ncpoly> <ncpoly>a1*a2*a1*a2-1</ncpoly> <ncpoly>a1*a3*a1*a3-1</ncpoly> <ncpoly>a1*a4*a1*a4-1</ncpoly> <ncpoly>a2*a1*a2*a1-1</ncpoly> <ncpoly>a2*a3*a2*a3-1</ncpoly> <ncpoly>a2*a4*a2*a4-1</ncpoly> <ncpoly>a3*a1*a3*a1-1</ncpoly> <ncpoly>a3*a2*a3*a2-1</ncpoly> <ncpoly>a3*a4*a3*a4-1</ncpoly> <ncpoly>a4*a1*a4*a1-1</ncpoly> <ncpoly>a4*a2*a4*a2-1</ncpoly> <ncpoly>a4*a3*a4*a3-1</ncpoly> </basis> <Comment>Alternating_group_4</Comment> </FREEALGEBRA>
Alternating group 5
<FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier"> <vars>a1,a2,a3,a4,a5</vars> <uptoDeg>5</uptoDeg> <basis> <ncpoly>a1^3-1</ncpoly> <ncpoly>a2^3-1</ncpoly> <ncpoly>a3^3-1</ncpoly> <ncpoly>a4^3-1</ncpoly> <ncpoly>a5^3-1</ncpoly> <ncpoly>a1*a2*a1*a2-1</ncpoly> <ncpoly>a1*a3*a1*a3-1</ncpoly> <ncpoly>a1*a4*a1*a4-1</ncpoly> <ncpoly>a1*a5*a1*a5-1</ncpoly> <ncpoly>a2*a1*a2*a1-1</ncpoly> <ncpoly>a2*a3*a2*a3-1</ncpoly> <ncpoly>a2*a4*a2*a4-1</ncpoly> <ncpoly>a2*a5*a2*a5-1</ncpoly> <ncpoly>a3*a1*a3*a1-1</ncpoly> <ncpoly>a3*a2*a3*a2-1</ncpoly> <ncpoly>a3*a4*a3*a4-1</ncpoly> <ncpoly>a3*a5*a3*a5-1</ncpoly> <ncpoly>a4*a1*a4*a1-1</ncpoly> <ncpoly>a4*a2*a4*a2-1</ncpoly> <ncpoly>a4*a3*a4*a3-1</ncpoly> <ncpoly>a4*a5*a4*a5-1</ncpoly> <ncpoly>a5*a1*a5*a1-1</ncpoly> <ncpoly>a5*a2*a5*a2-1</ncpoly> <ncpoly>a5*a3*a5*a3-1</ncpoly> <ncpoly>a5*a4*a5*a4-1</ncpoly> </basis> <Comment>Alternating_group_5</Comment> </FREEALGEBRA>
Alternating group 6
<FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier"> <vars>a1,a2,a3,a4,a5,a6</vars> <uptoDeg>5</uptoDeg> <basis> <ncpoly>a1^3-1</ncpoly> <ncpoly>a2^3-1</ncpoly> <ncpoly>a3^3-1</ncpoly> <ncpoly>a4^3-1</ncpoly> <ncpoly>a5^3-1</ncpoly> <ncpoly>a6^3-1</ncpoly> <ncpoly>a1*a2*a1*a2-1</ncpoly> <ncpoly>a1*a3*a1*a3-1</ncpoly> <ncpoly>a1*a4*a1*a4-1</ncpoly> <ncpoly>a1*a5*a1*a5-1</ncpoly> <ncpoly>a1*a6*a1*a6-1</ncpoly> <ncpoly>a2*a1*a2*a1-1</ncpoly> <ncpoly>a2*a3*a2*a3-1</ncpoly> <ncpoly>a2*a4*a2*a4-1</ncpoly> <ncpoly>a2*a5*a2*a5-1</ncpoly> <ncpoly>a2*a6*a2*a6-1</ncpoly> <ncpoly>a3*a1*a3*a1-1</ncpoly> <ncpoly>a3*a2*a3*a2-1</ncpoly> <ncpoly>a3*a4*a3*a4-1</ncpoly> <ncpoly>a3*a5*a3*a5-1</ncpoly> <ncpoly>a3*a6*a3*a6-1</ncpoly> <ncpoly>a4*a1*a4*a1-1</ncpoly> <ncpoly>a4*a2*a4*a2-1</ncpoly> <ncpoly>a4*a3*a4*a3-1</ncpoly> <ncpoly>a4*a5*a4*a5-1</ncpoly> <ncpoly>a4*a6*a4*a6-1</ncpoly> <ncpoly>a5*a1*a5*a1-1</ncpoly> <ncpoly>a5*a2*a5*a2-1</ncpoly> <ncpoly>a5*a3*a5*a3-1</ncpoly> <ncpoly>a5*a4*a5*a4-1</ncpoly> <ncpoly>a5*a6*a5*a6-1</ncpoly> <ncpoly>a6*a1*a6*a1-1</ncpoly> <ncpoly>a6*a2*a6*a2-1</ncpoly> <ncpoly>a6*a3*a6*a3-1</ncpoly> <ncpoly>a6*a4*a6*a4-1</ncpoly> <ncpoly>a6*a5*a6*a5-1</ncpoly> </basis> <Comment>Alternating_group_6</Comment> </FREEALGEBRA>