ApCoCoA-1:Free groups
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Description
The relations of a free group with n generators only consists of the existence of the invers elements. Any element of a free group has a unique representation.
F(n) = <a_{1},...,a_{n} | a_{i}a_{i}^{-1} = a_{i}^{-1}a_{i} = 1>
Reference
Kharlampovich, Olga; Myasnikov, Alexei, "Elementary theory of free non-abelian groups". J. Algebra 302 (2): 451–552, 2006.
Computation
/*Use the ApCoCoA package ncpoly.*/ // Number of free group MEMORY.N:=4; Use ZZ/(2)[x[1..MEMORY.N],y[1..MEMORY.N]]; NC.SetOrdering("LLEX"); Define CreateRelationsFree() Relations:=[]; For Index1 := 1 To MEMORY.N Do Append(Relations,[[x[Index1],y[Index1]],[1]]); Append(Relations,[[y[Index1],x[Index1]],[1]]); EndFor; Return Relations; EndDefine; Relations:=CreateRelationsFree(); Relations; Gb:=NC.GB(Relations); Gb;
Examples in Symbolic Data Format
Free group 4
<FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> <vars>x1,x2,x3,x4,y1,y2,y3,y4</vars> <basis> <ncpoly>x1*y1-1</ncpoly> <ncpoly>y1*x1-1</ncpoly> <ncpoly>x2*y2-1</ncpoly> <ncpoly>y2*x2-1</ncpoly> <ncpoly>x3*y3-1</ncpoly> <ncpoly>y3*x3-1</ncpoly> <ncpoly>x4*y4-1</ncpoly> <ncpoly>y4*x4-1</ncpoly> </basis> <Comment>Free_group_4</Comment> </FREEALGEBRA>
Free group 5
<FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> <vars>x1,x2,x3,x4,x5,y1,y2,y3,y4,y5</vars> <basis> <ncpoly>x1*y1-1</ncpoly> <ncpoly>y1*x1-1</ncpoly> <ncpoly>x2*y2-1</ncpoly> <ncpoly>y2*x2-1</ncpoly> <ncpoly>x3*y3-1</ncpoly> <ncpoly>y3*x3-1</ncpoly> <ncpoly>x4*y4-1</ncpoly> <ncpoly>y4*x4-1</ncpoly> <ncpoly>x5*y5-1</ncpoly> <ncpoly>y5*x5-1</ncpoly> </basis> <Comment>Free_group_5</Comment> </FREEALGEBRA>
Free group 6
<FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> <vars>x1,x2,x3,x4,x5,x6,y1,y2,y3,y4,y5,y6</vars> <basis> <ncpoly>x1*y1-1</ncpoly> <ncpoly>y1*x1-1</ncpoly> <ncpoly>x2*y2-1</ncpoly> <ncpoly>y2*x2-1</ncpoly> <ncpoly>x3*y3-1</ncpoly> <ncpoly>y3*x3-1</ncpoly> <ncpoly>x4*y4-1</ncpoly> <ncpoly>y4*x4-1</ncpoly> <ncpoly>x5*y5-1</ncpoly> <ncpoly>y5*x5-1</ncpoly> <ncpoly>x6*y6-1</ncpoly> <ncpoly>y6*x6-1</ncpoly> </basis> <Comment>Free_group_6</Comment> </FREEALGEBRA>
Free group 7
<FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> <vars>x1,x2,x3,x4,x5,x6,x7,y1,y2,y3,y4,y5,y6,y7</vars> <basis> <ncpoly>x1*y1-1</ncpoly> <ncpoly>y1*x1-1</ncpoly> <ncpoly>x2*y2-1</ncpoly> <ncpoly>y2*x2-1</ncpoly> <ncpoly>x3*y3-1</ncpoly> <ncpoly>y3*x3-1</ncpoly> <ncpoly>x4*y4-1</ncpoly> <ncpoly>y4*x4-1</ncpoly> <ncpoly>x5*y5-1</ncpoly> <ncpoly>y5*x5-1</ncpoly> <ncpoly>x6*y6-1</ncpoly> <ncpoly>y6*x6-1</ncpoly> <ncpoly>x7*y7-1</ncpoly> <ncpoly>y7*x7-1</ncpoly> </basis> <Comment>Free_group_7</Comment> </FREEALGEBRA>
Free group 8
<FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> <vars>x1,x2,x3,x4,x5,x6,x7,x8,y1,y2,y3,y4,y5,y6,y7,y8</vars> <basis> <ncpoly>x1*y1-1</ncpoly> <ncpoly>y1*x1-1</ncpoly> <ncpoly>x2*y2-1</ncpoly> <ncpoly>y2*x2-1</ncpoly> <ncpoly>x3*y3-1</ncpoly> <ncpoly>y3*x3-1</ncpoly> <ncpoly>x4*y4-1</ncpoly> <ncpoly>y4*x4-1</ncpoly> <ncpoly>x5*y5-1</ncpoly> <ncpoly>y5*x5-1</ncpoly> <ncpoly>x6*y6-1</ncpoly> <ncpoly>y6*x6-1</ncpoly> <ncpoly>x7*y7-1</ncpoly> <ncpoly>y7*x7-1</ncpoly> <ncpoly>x8*y8-1</ncpoly> <ncpoly>y8*x8-1</ncpoly> </basis> <Comment>Free_group_8</Comment> </FREEALGEBRA>
Free group 9
<FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> <vars>x1,x2,x3,x4,x5,x6,x7,x8,x9,y1,y2,y3,y4,y5,y6,y7,y8,y9</vars> <basis> <ncpoly>x1*y1-1</ncpoly> <ncpoly>y1*x1-1</ncpoly> <ncpoly>x2*y2-1</ncpoly> <ncpoly>y2*x2-1</ncpoly> <ncpoly>x3*y3-1</ncpoly> <ncpoly>y3*x3-1</ncpoly> <ncpoly>x4*y4-1</ncpoly> <ncpoly>y4*x4-1</ncpoly> <ncpoly>x5*y5-1</ncpoly> <ncpoly>y5*x5-1</ncpoly> <ncpoly>x6*y6-1</ncpoly> <ncpoly>y6*x6-1</ncpoly> <ncpoly>x7*y7-1</ncpoly> <ncpoly>y7*x7-1</ncpoly> <ncpoly>x8*y8-1</ncpoly> <ncpoly>y8*x8-1</ncpoly> <ncpoly>x9*y9-1</ncpoly> <ncpoly>y9*x9-1</ncpoly> </basis> <Comment>Free_group_9</Comment> </FREEALGEBRA>
Free group 10
<FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> <vars>x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,y1,y2,y3,y4,y5,y6,y7,y8,y9,y10</vars> <basis> <ncpoly>x1*y1-1</ncpoly> <ncpoly>y1*x1-1</ncpoly> <ncpoly>x2*y2-1</ncpoly> <ncpoly>y2*x2-1</ncpoly> <ncpoly>x3*y3-1</ncpoly> <ncpoly>y3*x3-1</ncpoly> <ncpoly>x4*y4-1</ncpoly> <ncpoly>y4*x4-1</ncpoly> <ncpoly>x5*y5-1</ncpoly> <ncpoly>y5*x5-1</ncpoly> <ncpoly>x6*y6-1</ncpoly> <ncpoly>y6*x6-1</ncpoly> <ncpoly>x7*y7-1</ncpoly> <ncpoly>y7*x7-1</ncpoly> <ncpoly>x8*y8-1</ncpoly> <ncpoly>y8*x8-1</ncpoly> <ncpoly>x9*y9-1</ncpoly> <ncpoly>y9*x9-1</ncpoly> <ncpoly>x10*y10-1</ncpoly> <ncpoly>y10*x10-1</ncpoly> </basis> <Comment>Free_group_10</Comment> </FREEALGEBRA>
Free group 11
<FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> <vars>x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11</vars> <basis> <ncpoly>x1*y1-1</ncpoly> <ncpoly>y1*x1-1</ncpoly> <ncpoly>x2*y2-1</ncpoly> <ncpoly>y2*x2-1</ncpoly> <ncpoly>x3*y3-1</ncpoly> <ncpoly>y3*x3-1</ncpoly> <ncpoly>x4*y4-1</ncpoly> <ncpoly>y4*x4-1</ncpoly> <ncpoly>x5*y5-1</ncpoly> <ncpoly>y5*x5-1</ncpoly> <ncpoly>x6*y6-1</ncpoly> <ncpoly>y6*x6-1</ncpoly> <ncpoly>x7*y7-1</ncpoly> <ncpoly>y7*x7-1</ncpoly> <ncpoly>x8*y8-1</ncpoly> <ncpoly>y8*x8-1</ncpoly> <ncpoly>x9*y9-1</ncpoly> <ncpoly>y9*x9-1</ncpoly> <ncpoly>x10*y10-1</ncpoly> <ncpoly>y10*x10-1</ncpoly> <ncpoly>x11*y11-1</ncpoly> <ncpoly>y11*x11-1</ncpoly> </basis> <Comment>Free_group_11</Comment> </FREEALGEBRA>