Difference between revisions of "ApCoCoA-1:FreeAbelian groups"
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(New page: === <div id="FreeAbelian_groups">Free abelian groups</div> === ==== Description ==== Every element in a free abelian group can be written in o...) |
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=== <div id="FreeAbelian_groups">[[:ApCoCoA:Symbolic data#Free_abelian_group|Free abelian groups]]</div> === | === <div id="FreeAbelian_groups">[[:ApCoCoA:Symbolic data#Free_abelian_group|Free abelian groups]]</div> === | ||
==== Description ==== | ==== Description ==== | ||
| − | Every element in a free abelian group can be written in only way as a finite linear combination. | + | 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: | 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> | ||
Revision as of 08:01, 16 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 (1970). Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press.)
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;
