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

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>

Cyclic group 7

 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>a</vars>
 	<basis>
 	<ncpoly>a^7-1</ncpoly>
 	</basis>
 	<Comment>Cyclic_group_7</Comment>
 </FREEALGEBRA> 

Cyclic group 8

 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>a</vars>
 	<basis>
 	<ncpoly>a^8-1</ncpoly>
 	</basis>
 	<Comment>Cyclic_group_8</Comment>
 </FREEALGEBRA> 

Cyclic group 9

 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>a</vars>
 	<basis>
 	<ncpoly>a^9-1</ncpoly>
 	</basis>
 	<Comment>Cyclic_group_9</Comment>
 </FREEALGEBRA> 

Cyclic group 10

 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>a</vars>
 	<basis>
 	<ncpoly>a^(10)-1</ncpoly>
 	</basis>
 	<Comment>Cyclic_group_10</Comment>
 </FREEALGEBRA> 

Cyclic group 11

 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>a</vars>
 	<basis>
 	<ncpoly>a^(11)-1</ncpoly>
 	</basis>
 	<Comment>Cyclic_group_11</Comment>
 </FREEALGEBRA> 

Cyclic group 12

 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>a</vars>
 	<basis>
 	<ncpoly>a^(12)-1</ncpoly>
 	</basis>
 	<Comment>Cyclic_group_12</Comment>
 </FREEALGEBRA> 

Cyclic group 13

 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>a</vars>
 	<basis>
 	<ncpoly>a^(13)-1</ncpoly>
 	</basis>
 	<Comment>Cyclic_group_13</Comment>
 </FREEALGEBRA> 

Cyclic group 14

 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>a</vars>
 	<basis>
 	<ncpoly>a^(14)-1</ncpoly>
 	</basis>
 	<Comment>Cyclic_group_14</Comment>
 </FREEALGEBRA> 

Cyclic group 15

 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>a</vars>
 	<basis>
 	<ncpoly>a^(15)-1</ncpoly>
 	</basis>
 	<Comment>Cyclic_group_15</Comment>
 </FREEALGEBRA> 

Cyclic group 16

 <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
 	<vars>a</vars>
 	<basis>
 	<ncpoly>a^(16)-1</ncpoly>
 	</basis>
 	<Comment>Cyclic_group_16</Comment>
 </FREEALGEBRA> 

Cyclic group 17

 <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
 	<vars>a</vars>
 	<basis>
 	<ncpoly>a^(17)-1</ncpoly>
 	</basis>
 	<Comment>Cyclic_group_17</Comment>
 </FREEALGEBRA> 

Cyclic group 18

 <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
 	<vars>a</vars>
 	<basis>
 	<ncpoly>a^(18)-1</ncpoly>
 	</basis>
 	<Comment>Cyclic_group_18</Comment>
 </FREEALGEBRA> 

Cyclic group 19

 <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
 	<vars>a</vars>
 	<basis>
 	<ncpoly>a^(19)-1</ncpoly>
 	</basis>
 	<Comment>Cyclic_group_19</Comment>
 </FREEALGEBRA> 

Cyclic group 20

 <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
 	<vars>a</vars>
 	<basis>
 	<ncpoly>a^(20)-1</ncpoly>
 	</basis>
 	<Comment>Cyclic_group_20</Comment>
 </FREEALGEBRA> 

Cyclic group 21

 <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
 	<vars>a</vars>
 	<basis>
 	<ncpoly>a^(21)-1</ncpoly>
 	</basis>
 	<Comment>Cyclic_group_21</Comment>
 </FREEALGEBRA> 

Cyclic group 22

 <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
 	<vars>a</vars>
 	<basis>
 	<ncpoly>a^(22)-1</ncpoly>
 	</basis>
 	<Comment>Cyclic_group_22</Comment>
 </FREEALGEBRA> 

Cyclic group 23

 <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
 	<vars>a</vars>
 	<basis>
 	<ncpoly>a^(23)-1</ncpoly>
 	</basis>
 	<Comment>Cyclic_group_23</Comment>
 </FREEALGEBRA>