ApCoCoA-1:Cyclic groups: Difference between revisions
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=== <div id="Cyclic_groups">[[:ApCoCoA:Symbolic data#Cyclic_groups|Cyclic | === <div id="Cyclic_groups">[[:ApCoCoA:Symbolic data#Cyclic_groups|Cyclic Groups]]</div> === | ||
==== Description ==== | ==== 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 | 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 | ||
| Line 5: | Line 5: | ||
C(n) = <a | a^{n} = 1> | C(n) = <a | a^{n} = 1> | ||
==== Reference ==== | |||
Joseph A. Gallian, Contemporary Abstract Algebra (4th ed.), Boston: Houghton Mifflin, Chapter 4, 1998. | |||
==== Computation ==== | ==== Computation ==== | ||
| Line 16: | Line 17: | ||
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]]] | |||
==== Example in Symbolic Data Format ==== | |||
<FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> | |||
<vars>a</vars> | |||
<basis> | |||
<ncpoly>a^5-1</ncpoly> | |||
</basis> | |||
<Comment>Cyclic_group_5</Comment> | |||
</FREEALGEBRA> | |||
Latest revision as of 20:28, 22 April 2014
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
Joseph A. Gallian, Contemporary Abstract Algebra (4th ed.), Boston: Houghton Mifflin, Chapter 4, 1998.
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]]]
Example in Symbolic Data Format
<FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> <vars>a</vars> <basis> <ncpoly>a^5-1</ncpoly> </basis> <Comment>Cyclic_group_5</Comment> </FREEALGEBRA>
