Difference between revisions of "ApCoCoA-1:Cyclic groups"
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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 | ||
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// RESULT for MEMORY.N = 5 :: [[[a^5], [1]]] | // RESULT for MEMORY.N = 5 :: [[[a^5], [1]]] | ||
| − | ==== | + | ==== Example in Symbolic Data Format ==== |
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<FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> | <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> | ||
<vars>a</vars> | <vars>a</vars> | ||
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</basis> | </basis> | ||
<Comment>Cyclic_group_5</Comment> | <Comment>Cyclic_group_5</Comment> | ||
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</FREEALGEBRA> | </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>
