ApCoCoA-1:Cyclic groups

Cyclic groups

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