Difference between revisions of "ApCoCoA-1:FreeAbelian groups"
From ApCoCoAWiki
| Line 4: | Line 4: | ||
following: | following: | ||
Z(n) = <a_{1},...,a_{n} | [a_{i},a_{j}] = 1 for all i,j> | Z(n) = <a_{1},...,a_{n} | [a_{i},a_{j}] = 1 for all i,j> | ||
| − | + | ||
| + | ==== Reference ==== | ||
| + | Phillip A. Griffith, Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press, 1970. | ||
| + | |||
==== Computation ==== | ==== Computation ==== | ||
/*Use the ApCoCoA package ncpoly.*/ | /*Use the ApCoCoA package ncpoly.*/ | ||
// Number of free abelian group | // Number of free abelian group | ||
| − | |||
MEMORY.N:=3; | MEMORY.N:=3; | ||
Use ZZ/(2)[x[1..MEMORY.N],y[1..MEMORY.N]]; | Use ZZ/(2)[x[1..MEMORY.N],y[1..MEMORY.N]]; | ||
NC.SetOrdering("LLEX"); | NC.SetOrdering("LLEX"); | ||
| + | |||
Define CreateRelationsFreeAbelian() | Define CreateRelationsFreeAbelian() | ||
Relations:=[]; | Relations:=[]; | ||
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EndFor; | EndFor; | ||
EndFor; | EndFor; | ||
| − | |||
| − | |||
Return Relations; | Return Relations; | ||
EndDefine; | EndDefine; | ||
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Relations:=CreateRelationsFreeAbelian(); | Relations:=CreateRelationsFreeAbelian(); | ||
Relations; | Relations; | ||
| + | |||
GB:=NC.GB(Relations); | GB:=NC.GB(Relations); | ||
GB; | GB; | ||
Revision as of 09:14, 23 August 2013
Description
Every element in a free abelian group can be written in only way as a finite linear combination. A representation is given by the following:
Z(n) = <a_{1},...,a_{n} | [a_{i},a_{j}] = 1 for all i,j>
Reference
Phillip A. Griffith, Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press, 1970.
Computation
/*Use the ApCoCoA package ncpoly.*/
// Number of free abelian group
MEMORY.N:=3;
Use ZZ/(2)[x[1..MEMORY.N],y[1..MEMORY.N]];
NC.SetOrdering("LLEX");
Define CreateRelationsFreeAbelian()
Relations:=[];
For Index1 := 1 To MEMORY.N Do
For Index2 := 1 To MEMORY.N Do
Append(Relations,[[x[Index1],x[Index2],y[Index1],y[Index2]],[1]]);
EndFor;
EndFor;
Return Relations;
EndDefine;
Relations:=CreateRelationsFreeAbelian();
Relations;
GB:=NC.GB(Relations);
GB;
