Difference between revisions of "ApCoCoA-1:Cyclic groups"
From ApCoCoAWiki
(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...) |
|||
| Line 14: | Line 14: | ||
MEMORY.N:=5; | MEMORY.N:=5; | ||
| − | Use ZZ/(2)[a | + | Use ZZ/(2)[a]; |
NC.SetOrdering("LLEX"); | NC.SetOrdering("LLEX"); | ||
Define CreateRelationsCyclic() | Define CreateRelationsCyclic() | ||
Relations:=[]; | Relations:=[]; | ||
| − | |||
| − | |||
| − | |||
// add relation a^n = 1 | // add relation a^n = 1 | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
Append(Relations,[[a^MEMORY.N],[-1]]); | Append(Relations,[[a^MEMORY.N],[-1]]); | ||
| − | |||
Return Relations; | Return Relations; | ||
EndDefine; | EndDefine; | ||
Revision as of 14:49, 12 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: 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];
NC.SetOrdering("LLEX");
Define CreateRelationsCyclic()
Relations:=[];
// add relation a^n = 1
Append(Relations,[[a^MEMORY.N],[-1]]);
Return Relations;
EndDefine;
Relations:=CreateRelationsCyclic();
// Compute the Groebner Bases
