# 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 |

## 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>