Difference between revisions of "ApCoCoA-1:SpecialLinearPrime group"
From ApCoCoAWiki
StrohmeierB (talk | contribs) (New page: === <div id="special_linear_group with prime p">Special Linear Group with prime p</div> === ==== Description ==== For p is prim...) |
StrohmeierB (talk | contribs) |
||
| (2 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | === <div id="special_linear_group with prime p">[[:ApCoCoA:Symbolic data#special_linear_group with prime p|Special Linear Group with | + | === <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: | ||
| Line 7: | Line 7: | ||
not found yet | not found yet | ||
==== Computation ==== | ==== 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==== | ====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> | ||
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>
