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...) |
StrohmeierB (talk | contribs) |
||
| (20 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
| − | === <div id="Cyclic_groups">[[:ApCoCoA:Symbolic data#Cyclic_groups|Cyclic | + | === <div id="Cyclic_groups">[[:ApCoCoA:Symbolic data#Cyclic_groups|Cyclic Groups]]</div> === |
==== Description ==== | ==== 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 | 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 | ||
| Line 5: | Line 5: | ||
C(n) = <a | a^{n} = 1> | C(n) = <a | a^{n} = 1> | ||
| − | + | ==== Reference ==== | |
| + | Joseph A. Gallian, Contemporary Abstract Algebra (4th ed.), Boston: Houghton Mifflin, Chapter 4, 1998. | ||
==== Computation ==== | ==== Computation ==== | ||
| Line 14: | Line 15: | ||
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:=[]; | |
| − | + | // Add relation a^n = 1 | |
| − | + | Append(Relations,[[a^MEMORY.N],[1]]); | |
| − | + | Return Relations; | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
EndDefine; | EndDefine; | ||
Relations:=CreateRelationsCyclic(); | Relations:=CreateRelationsCyclic(); | ||
| + | Relations; | ||
| − | // Compute | + | // Compute a Groebner Basis. |
| + | Gb:=NC.GB(Relations); | ||
| + | Gb; | ||
| + | // RESULT for MEMORY.N = 5 :: [[[a^5], [1]]] | ||
| + | |||
| + | ==== Example in Symbolic Data Format ==== | ||
| + | <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> | ||
| + | <vars>a</vars> | ||
| + | <basis> | ||
| + | <ncpoly>a^5-1</ncpoly> | ||
| + | </basis> | ||
| + | <Comment>Cyclic_group_5</Comment> | ||
| + | </FREEALGEBRA> | ||
Latest revision as of 20:28, 22 April 2014
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]]]
Example in Symbolic Data Format
<FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> <vars>a</vars> <basis> <ncpoly>a^5-1</ncpoly> </basis> <Comment>Cyclic_group_5</Comment> </FREEALGEBRA>
