Difference between revisions of "ApCoCoA-1:Cyclic groups"
From ApCoCoAWiki
| Line 21: | Line 21: | ||
Relations:=[]; | Relations:=[]; | ||
// Add relation a^n = 1 | // Add relation a^n = 1 | ||
| − | Append(Relations,[[a^MEMORY.N],[ | + | Append(Relations,[[a^MEMORY.N],[1]]); |
Return Relations; | Return Relations; | ||
EndDefine; | EndDefine; | ||
Revision as of 07:35, 23 August 2013
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]]]
