# ApCoCoA-1:Cyclic groups

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Revision as of 14:25, 12 August 2013 by F lorenz (talk | contribs) (New page: === <div id="Cyclic_groups">Cyclic groups</div> === ==== Description ==== Every cyclic group is generated by a single element a. If n is finite the...)

#### 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,b]; NC.SetOrdering("LLEX"); Define CreateRelationsCyclic() Relations:=[]; // add relation of invers elements a and b Append(Relations,[[a,b],[-1]]); // add relation a^n = 1 BufferA:=[]; For Index1:=1 To MEMORY.N Do Append(BufferA,a); EndFor; Append(Relations,[[BufferA],[-1]]); /* Alternative Implementation Append(Relations,[[a^MEMORY.N],[-1]]); */ Return Relations; EndDefine; Relations:=CreateRelationsCyclic(); // Compute the Groebner Bases