Difference between revisions of "ApCoCoA-1:Cyclic groups"

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=== <div id="Cyclic_groups">[[:ApCoCoA:Symbolic data#Cyclic_groups|Cyclic groups]]</div> ===
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=== <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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==== Reference ====
 
==== Reference ====
Gallian, Joseph (1998), Contemporary abstract algebra (4th ed.), Boston: Houghton Mifflin, Chapter 4.
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Joseph A. Gallian, Contemporary Abstract Algebra (4th ed.), Boston: Houghton Mifflin, Chapter 4, 1998.
  
 
==== Computation ====
 
==== Computation ====
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     Relations:=[];
 
     Relations:=[];
 
     // Add relation a^n = 1
 
     // Add relation a^n = 1
     Append(Relations,[[a^MEMORY.N],[-1]]);
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     Append(Relations,[[a^MEMORY.N],[1]]);
 
     Return Relations;
 
     Return Relations;
 
   EndDefine;
 
   EndDefine;
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   Gb;
 
   Gb;
 
   // RESULT for MEMORY.N = 5 :: [[[a^5], [1]]]
 
   // RESULT for MEMORY.N = 5 :: [[[a^5], [1]]]
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==== Example in Symbolic Data Format ====
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  <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
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  <vars>a</vars>
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  <basis>
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  <ncpoly>a^5-1</ncpoly>
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  </basis>
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  <Comment>Cyclic_group_5</Comment>
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  </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>