Difference between revisions of "ApCoCoA-1:Other1 groups"
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| − | === <div id="Other_groups">[[:ApCoCoA:Symbolic data#Other_groups|Other | + | === <div id="Other_groups">[[:ApCoCoA:Symbolic data#Other_groups|Other Groups]]</div> === |
==== Description ==== | ==== Description ==== | ||
This group has the following representation: | This group has the following representation: | ||
| Line 11: | Line 11: | ||
/*Use the ApCoCoA package ncpoly.*/ | /*Use the ApCoCoA package ncpoly.*/ | ||
| + | //K is congruent to 3 mod 6 | ||
| + | MEMORY.K:=3; | ||
// a is invers to c and b is invers to d | // a is invers to c and b is invers to d | ||
Use ZZ/(2)[a,b,c,d]; | Use ZZ/(2)[a,b,c,d]; | ||
| Line 26: | Line 28: | ||
Append(Relations,[[a,a,d,d,d,d,d,d],[1]]); | Append(Relations,[[a,a,d,d,d,d,d,d],[1]]); | ||
| − | // add the relation (ab^{-1})^{3}ab^{-2}ab^{k}a^{-1}b = 1 | + | // add the relation (ab^{-1})^{3}ab^{-2}ab^{k}a^{-1}b = 1 |
| − | Append(Relations,[[a,d,a,d,a,d,a,d,d,a,b | + | Append(Relations,[[a,d,a,d,a,d,a,d,d,a,b^MEMORY.K,c,b],[1]]); |
| + | |||
Return Relations; | Return Relations; | ||
EndDefine; | EndDefine; | ||
Relations:=CreateRelationsOther1(); | Relations:=CreateRelationsOther1(); | ||
| − | + | Gb:=NC.GB(Relations,31,1,100,1000); | |
| + | ====Examples in Symbolic Data Format==== | ||
| + | =====Other group 1 k=3===== | ||
| + | <FREEALGEBRA createdAt="2014-01-27" createdBy="strohmeier"> | ||
| + | <vars>a,b,c,d</vars> | ||
| + | <uptoDeg>10</uptoDeg> | ||
| + | <basis> | ||
| + | <ncpoly>a*c-1</ncpoly> | ||
| + | <ncpoly>c*a-1</ncpoly> | ||
| + | <ncpoly>b*d-1</ncpoly> | ||
| + | <ncpoly>d*b-1</ncpoly> | ||
| + | <ncpoly>(a^2)*(d^6)-1</ncpoly> | ||
| + | <ncpoly>((a*d)^3)*a*d*d*a*(b^3)*c*b-1</ncpoly> | ||
| + | </basis> | ||
| + | <Comment>Other_groups1k3</Comment> | ||
| + | </FREEALGEBRA> | ||
| + | =====Other group 1 k=39===== | ||
| + | <FREEALGEBRA createdAt="2014-01-27" createdBy="strohmeier"> | ||
| + | <vars>a,b,c,d</vars> | ||
| + | <uptoDeg>10</uptoDeg> | ||
| + | <basis> | ||
| + | <ncpoly>a*c-1</ncpoly> | ||
| + | <ncpoly>c*a-1</ncpoly> | ||
| + | <ncpoly>b*d-1</ncpoly> | ||
| + | <ncpoly>d*b-1</ncpoly> | ||
| + | <ncpoly>(a^2)*(d^6)-1</ncpoly> | ||
| + | <ncpoly>((a*d)^3)*a*d*d*a*(b^39)*c*b-1</ncpoly> | ||
| + | </basis> | ||
| + | <Comment>The partial LLex Gb has 234 elements</Comment> | ||
| + | <Comment>Other_groups1k39</Comment> | ||
| + | </FREEALGEBRA> | ||
| + | |||
| + | =====Other group 1 k=63===== | ||
| + | <FREEALGEBRA createdAt="2014-01-27" createdBy="strohmeier"> | ||
| + | <vars>a,b,c,d</vars> | ||
| + | <uptoDeg>9</uptoDeg> | ||
| + | <basis> | ||
| + | <ncpoly>a*c-1</ncpoly> | ||
| + | <ncpoly>c*a-1</ncpoly> | ||
| + | <ncpoly>b*d-1</ncpoly> | ||
| + | <ncpoly>d*b-1</ncpoly> | ||
| + | <ncpoly>(a^2)*(d^6)-1</ncpoly> | ||
| + | <ncpoly>((a*d)^3)*a*d*d*a*(b^63)*c*b-1</ncpoly> | ||
| + | </basis> | ||
| + | <Comment>The partial LLex Gb has 137 elements</Comment> | ||
| + | <Comment>Other_groups1k63</Comment> | ||
| + | </FREEALGEBRA> | ||
Latest revision as of 21:08, 22 April 2014
Description
This group has the following representation:
G = <a,b | a^{2}b^{-6} = (ab^{-1})^{3}ab^{-2}ab^{k}a^{-1}b = 1>
where k is congruent to 3 mod 6.
Reference
No reference available
Computation
/*Use the ApCoCoA package ncpoly.*/
//K is congruent to 3 mod 6
MEMORY.K:=3;
// a is invers to c and b is invers to d
Use ZZ/(2)[a,b,c,d];
NC.SetOrdering("LLEX");
Define CreateRelationsOther1()
Relations:=[];
// add the invers relations ac = ca = bd = db = 1
Append(Relations,[[a,c],[1]]);
Append(Relations,[[c,a],[1]]);
Append(Relations,[[b,d],[1]]);
Append(Relations,[[d,b],[1]]);
// add the relation a^{2}b^{-6} = aadddddd = 1
Append(Relations,[[a,a,d,d,d,d,d,d],[1]]);
// add the relation (ab^{-1})^{3}ab^{-2}ab^{k}a^{-1}b = 1
Append(Relations,[[a,d,a,d,a,d,a,d,d,a,b^MEMORY.K,c,b],[1]]);
Return Relations;
EndDefine;
Relations:=CreateRelationsOther1();
Gb:=NC.GB(Relations,31,1,100,1000);
Examples in Symbolic Data Format
Other group 1 k=3
<FREEALGEBRA createdAt="2014-01-27" createdBy="strohmeier"> <vars>a,b,c,d</vars> <uptoDeg>10</uptoDeg> <basis> <ncpoly>a*c-1</ncpoly> <ncpoly>c*a-1</ncpoly> <ncpoly>b*d-1</ncpoly> <ncpoly>d*b-1</ncpoly> <ncpoly>(a^2)*(d^6)-1</ncpoly> <ncpoly>((a*d)^3)*a*d*d*a*(b^3)*c*b-1</ncpoly> </basis> <Comment>Other_groups1k3</Comment> </FREEALGEBRA>
Other group 1 k=39
<FREEALGEBRA createdAt="2014-01-27" createdBy="strohmeier"> <vars>a,b,c,d</vars> <uptoDeg>10</uptoDeg> <basis> <ncpoly>a*c-1</ncpoly> <ncpoly>c*a-1</ncpoly> <ncpoly>b*d-1</ncpoly> <ncpoly>d*b-1</ncpoly> <ncpoly>(a^2)*(d^6)-1</ncpoly> <ncpoly>((a*d)^3)*a*d*d*a*(b^39)*c*b-1</ncpoly> </basis> <Comment>The partial LLex Gb has 234 elements</Comment> <Comment>Other_groups1k39</Comment> </FREEALGEBRA>
Other group 1 k=63
<FREEALGEBRA createdAt="2014-01-27" createdBy="strohmeier"> <vars>a,b,c,d</vars> <uptoDeg>9</uptoDeg> <basis> <ncpoly>a*c-1</ncpoly> <ncpoly>c*a-1</ncpoly> <ncpoly>b*d-1</ncpoly> <ncpoly>d*b-1</ncpoly> <ncpoly>(a^2)*(d^6)-1</ncpoly> <ncpoly>((a*d)^3)*a*d*d*a*(b^63)*c*b-1</ncpoly> </basis> <Comment>The partial LLex Gb has 137 elements</Comment> <Comment>Other_groups1k63</Comment> </FREEALGEBRA>
