Difference between revisions of "ApCoCoA-1:Alternating groups"
From ApCoCoAWiki
(New page: === <div id="Alternating_groups">Alternating groups</div> === ==== Description ==== The alternating groups is the group of all even permutatio...) |
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Use ZZ/(2)[a[1..MEMORY.N]]; | Use ZZ/(2)[a[1..MEMORY.N]]; | ||
NC.SetOrdering("LLEX"); | NC.SetOrdering("LLEX"); | ||
| + | |||
Define CreateRelationsAlternating() | Define CreateRelationsAlternating() | ||
Relations:=[]; | Relations:=[]; | ||
| Line 35: | Line 36: | ||
Relations:=CreateRelationsAlternating(); | Relations:=CreateRelationsAlternating(); | ||
| − | + | Gb:=NC.GB(Relations); | |
Revision as of 03:38, 22 September 2013
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);
