Difference between revisions of "ApCoCoA-1:Cyclic groups"
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// RESULT for MEMORY.N = 5 :: [[[a^5], [1]]] | // RESULT for MEMORY.N = 5 :: [[[a^5], [1]]] | ||
| − | ==== | + | ==== Examples in Symbolic Data format ==== |
=====Cyclic group 5===== | =====Cyclic group 5===== | ||
<FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> | <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> | ||
| Line 41: | Line 41: | ||
</basis> | </basis> | ||
<Comment>Cyclic_group_5</Comment> | <Comment>Cyclic_group_5</Comment> | ||
| + | </FREEALGEBRA> | ||
| + | =====Cyclic group 6===== | ||
| + | <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> | ||
| + | <vars>a</vars> | ||
| + | <basis> | ||
| + | <ncpoly>a^6-1</ncpoly> | ||
| + | </basis> | ||
| + | <Comment>Cyclic_group_6</Comment> | ||
</FREEALGEBRA> | </FREEALGEBRA> | ||
Revision as of 14:00, 6 March 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]]]
Examples in Symbolic Data format
Cyclic group 5
<FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> <vars>a</vars> <basis> <ncpoly>a^5-1</ncpoly> </basis> <Comment>Cyclic_group_5</Comment> </FREEALGEBRA>
Cyclic group 6
<FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
<vars>a</vars> <basis> <ncpoly>a^6-1</ncpoly> </basis> <Comment>Cyclic_group_6</Comment>
</FREEALGEBRA>
