ApCoCoA-1:Cyclic groups: Difference between revisions
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C(n) = <a | a^{n} = 1> | C(n) = <a | a^{n} = 1> | ||
==== Reference ==== | |||
Gallian, Joseph (1998), Contemporary abstract algebra (4th ed.), Boston: Houghton Mifflin, Chapter 4. | |||
==== Computation ==== | ==== Computation ==== | ||
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Use ZZ/(2)[a]; | Use ZZ/(2)[a]; | ||
NC.SetOrdering("LLEX"); | NC.SetOrdering("LLEX"); | ||
Define CreateRelationsCyclic() | Define CreateRelationsCyclic() | ||
Relations:=[]; | |||
// Add relation a^n = 1 | |||
Append(Relations,[[a^MEMORY.N],[-1]]); | |||
Return Relations; | |||
EndDefine; | EndDefine; | ||
Relations:=CreateRelationsCyclic(); | Relations:=CreateRelationsCyclic(); | ||
Relations; | |||
// Compute | // Compute a Groebner Basis. | ||
Gb:=NC.GB(Relations); | |||
Gb; | |||
// RESULT for MEMORY.N = 5 :: [[[a^5], [1]]] | // RESULT for MEMORY.N = 5 :: [[[a^5], [1]]] | ||
Revision as of 07:32, 23 August 2013
Description
Every cyclic group is generated by a single element a. If n is finite the group is isomorphic to Z/nZ, otherwise it can be interpreted as Z with the addition of integers as the group operation. For every cyclic group there only exists one subgroup containing a, the group itself.
C(n) = <a | a^{n} = 1>
Reference
Gallian, Joseph (1998), Contemporary abstract algebra (4th ed.), Boston: Houghton Mifflin, Chapter 4.
Computation
/*Use the ApCoCoA package ncpoly.*/
// Number of cyclic group
MEMORY.N:=5;
Use ZZ/(2)[a];
NC.SetOrdering("LLEX");
Define CreateRelationsCyclic()
Relations:=[];
// Add relation a^n = 1
Append(Relations,[[a^MEMORY.N],[-1]]);
Return Relations;
EndDefine;
Relations:=CreateRelationsCyclic();
Relations;
// Compute a Groebner Basis.
Gb:=NC.GB(Relations);
Gb;
// RESULT for MEMORY.N = 5 :: [[[a^5], [1]]]
