ApCoCoA-1:SpecialLinearPrime group
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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>