Difference between revisions of "ApCoCoA-1:Alternating groups"
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| − | === <div id="Alternating_groups">[[:ApCoCoA:Symbolic data#Alternating_groups|Alternating | + | === <div id="Alternating_groups">[[:ApCoCoA:Symbolic data#Alternating_groups|Alternating Groups]]</div> === |
==== Description ==== | ==== Description ==== | ||
The alternating groups is the group of all even permutations of a finite set. Every alternating group is a subgroups of | The alternating groups is the group of all even permutations of a finite set. Every alternating group is a subgroups of | ||
| Line 58: | Line 58: | ||
<FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier"> | <FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier"> | ||
<vars>a1,a2,a3,a4</vars> | <vars>a1,a2,a3,a4</vars> | ||
| + | <uptoDeg>6</uptoDeg> | ||
<basis> | <basis> | ||
<ncpoly>a1^3-1</ncpoly> | <ncpoly>a1^3-1</ncpoly> | ||
| Line 81: | Line 82: | ||
<FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier"> | <FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier"> | ||
<vars>a1,a2,a3,a4,a5</vars> | <vars>a1,a2,a3,a4,a5</vars> | ||
| + | <uptoDeg>5</uptoDeg> | ||
<basis> | <basis> | ||
<ncpoly>a1^3-1</ncpoly> | <ncpoly>a1^3-1</ncpoly> | ||
| Line 114: | Line 116: | ||
<FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier"> | <FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier"> | ||
<vars>a1,a2,a3,a4,a5,a6</vars> | <vars>a1,a2,a3,a4,a5,a6</vars> | ||
| − | <uptoDeg> | + | <uptoDeg>5</uptoDeg> |
<basis> | <basis> | ||
<ncpoly>a1^3-1</ncpoly> | <ncpoly>a1^3-1</ncpoly> | ||
Latest revision as of 21:04, 22 April 2014
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>
