# ApCoCoA-1:Cyclic groups

From ApCoCoAWiki

#### 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]]]