ApCoCoA-1:Alternating groups

Alternating groups

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);