Difference between revisions of "ApCoCoA-1:SpecialLinear group"

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(New page: === <div id="SpecialLinear_group">SpecialLinear group</div> === ==== Description ==== The special linear group over Z is the group of all n x...)
 
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=== <div id="SpecialLinear_group">[[:ApCoCoA:Symbolic data#SpecialLinear_group|SpecialLinear group]]</div> ===
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=== <div id="SpecialLinear_group">[[:ApCoCoA:Symbolic data#Special Linear_group|SpecialLinear group]]</div> ===
 
==== Description ====
 
==== Description ====
 
The special linear group over Z is the group of all n x n matrices with coefficients in Z which has a determinant equals 0.
 
The special linear group over Z is the group of all n x n matrices with coefficients in Z which has a determinant equals 0.

Revision as of 07:08, 10 September 2013

Description

The special linear group over Z is the group of all n x n matrices with coefficients in Z which has a determinant equals 0. A simple representation is given by:

 SL_2(Z) = <a,b | aba = bab, (aba)^4 = 1 >

Reference

Andrew Baker: An introduction to matrix groups and their applications

Computation

 /*Use the ApCoCoA package ncpoly.*/
 
 // a is invers to c, b is invers to d
 Use ZZ/(2)[a,b,c,d];
 NC.SetOrdering("LLEX");
 Define CreateRelationsSpecialLinear()
   Relations:=[];
   // add the invers relations
   Append(Relations,[[a,c],[1]]);
   Append(Relations,[[c,a],[1]]);
   Append(Relations,[[b,d],[1]]);
   Append(Relations,[[d,b],[1]]);
  
   // add the relation aba = bab 
   Append(Relations,[[a,b,a],[b,a,b]]);
   
   // add the relation (aba)^4 = 1
   Append(Relations,[[a,b,a,a,b,a,a,b,a,a,b,a],[1]]);
  
   Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsSpecialLinear();
 GB:=NC.GB(Relations,31,1,100,1000);