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

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(New page: === <div id="special_linear_group with prime p">Special Linear Group with prime p</div> === ==== Description ==== For p is prim...)
 
 
(2 intermediate revisions by the same user not shown)
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=== <div id="special_linear_group with prime p">[[:ApCoCoA:Symbolic data#special_linear_group with prime p|Special Linear Group with prime p]]</div> ===
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=== <div id="special_linear_group with prime p">[[:ApCoCoA:Symbolic data#special_linear_group with prime p|Special Linear Group with Prime p]]</div> ===
 
==== Description ====
 
==== Description ====
 
For p is prime has the Special Linear Group with prime p the following presentation:
 
For p is prime has the Special Linear Group with prime p the following presentation:
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not found yet
 
not found yet
 
==== Computation ====
 
==== Computation ====
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  /*Use the ApCoCoA package ncpoly.*/
 +
 +
// set the variables k,p,t
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// Note that p have to be prime
 +
MEMORY.K:=3;
 +
MEMORY.P:=2;
 +
MEMORY.T:=5;
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Use ZZ/(2)[x,y,a,b];
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NC.SetOrdering("LLEX");
 +
 +
  Define CreateRelationsSpeciallineargroupprime()
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  Relations:=[];
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    // add the inverse relations
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  Append(Relations,[[x,a],[1]]);
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  Append(Relations,[[a,x],[1]]);
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  Append(Relations,[[y,b],[1]]);
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  Append(Relations,[[b,y],[1]]);
 +
 
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  // add the relation x^2 = (xy)^3
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  Append(Relations,[[x,x],[x,y,x,y,x,y]]);
 +
 
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  // add the relation ((((x*y)^4)*x*y^t)^2)*(y^p)*(x^(2k))-1
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  Append(Relations,[[x,y,x,y,x,y,x,y,x,y^(MEMORY.T),x,y,x,y,x,y,x,y,x,y^(MEMORY.T),y^(MEMORY.P),x^(2*MEMORY.K)],[1]]);
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  Return Relations;
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EndDefine;
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Relations:=CreateRelationsSpeciallineargroupprime();
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Relations;
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Gb:=NC.GB(Relations,31,1,100,1000);
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Gb;
 +
 
====Example in Symbolic Data Format====
 
====Example in Symbolic Data Format====
 +
  <FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier">
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  <vars>x,y,a,b</vars>
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  <uptoDeg>12</uptoDeg>
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  <basis>
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  <ncpoly>x*x-(x*y)^3</ncpoly>
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  <ncpoly>((((x*y)^4)*x*y^5)^2)*(y^2)*(x^(2*3))-1</ncpoly>
 +
  </basis>
 +
  <Comment>The partial LLEX Gb has 285 elements</Comment>
 +
  <Comment>Special_Linear_group with prime p_k3p2t5</Comment>
 +
  </FREEALGEBRA>

Latest revision as of 21:03, 22 April 2014

Description

For p is prime has the Special Linear Group with prime p the following presentation:

  SL_2(p) = <x,y | x^{2} =(xy)^{3},(xy^{4}xy^{t})^{2}y^{p}x^{2k}=1>

Reference

not found yet

Computation

 /*Use the ApCoCoA package ncpoly.*/

// set the variables k,p,t
// Note that p have to be prime
MEMORY.K:=3;
MEMORY.P:=2;
MEMORY.T:=5;
Use ZZ/(2)[x,y,a,b];
NC.SetOrdering("LLEX");

 Define CreateRelationsSpeciallineargroupprime()
  Relations:=[];
  
   // add the inverse relations
  Append(Relations,[[x,a],[1]]);
  Append(Relations,[[a,x],[1]]);
  Append(Relations,[[y,b],[1]]);
  Append(Relations,[[b,y],[1]]);
  
  // add the relation x^2 = (xy)^3
  Append(Relations,[[x,x],[x,y,x,y,x,y]]);
  
  // add the relation ((((x*y)^4)*x*y^t)^2)*(y^p)*(x^(2k))-1
  Append(Relations,[[x,y,x,y,x,y,x,y,x,y^(MEMORY.T),x,y,x,y,x,y,x,y,x,y^(MEMORY.T),y^(MEMORY.P),x^(2*MEMORY.K)],[1]]);
  Return Relations;
EndDefine;

Relations:=CreateRelationsSpeciallineargroupprime();
Relations;

Gb:=NC.GB(Relations,31,1,100,1000);
Gb;

Example in Symbolic Data Format

 <FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier">
 	<vars>x,y,a,b</vars>
 	<uptoDeg>12</uptoDeg>
 	<basis>
 	<ncpoly>x*x-(x*y)^3</ncpoly>
 	<ncpoly>((((x*y)^4)*x*y^5)^2)*(y^2)*(x^(2*3))-1</ncpoly>
 	</basis>
 	<Comment>The partial LLEX Gb has 285 elements</Comment>
 	<Comment>Special_Linear_group with prime p_k3p2t5</Comment>
 </FREEALGEBRA>