ApCoCoA:Alternating groups

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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>