Difference between revisions of "ApCoCoA-1:Dicyclic groups"

From ApCoCoAWiki
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   Dic(n) = <a,b | a^{2n} = 1, a^{n} = b^{2}, b^{-1}ab = a^{-1}>
 
   Dic(n) = <a,b | a^{2n} = 1, a^{n} = b^{2}, b^{-1}ab = a^{-1}>
  
(Reference: Coxeter, H. S. M. (1974), "7.1 The Cyclic and Dicyclic groups", Regular Complex Polytopes, Cambridge University)
+
==== Reference ====
 +
Coxeter, H. S. M., "7.1 The Cyclic and Dicyclic groups", Regular Complex Polytopes, Cambridge University, 1974.
  
 
==== Computation ====
 
==== Computation ====
Line 13: Line 14:
 
   // Number of Dicyclic group (note that  the order is 4N)
 
   // Number of Dicyclic group (note that  the order is 4N)
 
   MEMORY.N:=5;
 
   MEMORY.N:=5;
 
 
 
    
 
    
 
   Use ZZ/(2)[a,b];
 
   Use ZZ/(2)[a,b];
 
   NC.SetOrdering("LLEX");
 
   NC.SetOrdering("LLEX");
 +
 
 
   Define CreateRelationsDicyclic()
 
   Define CreateRelationsDicyclic()
 
     Relations:=[];
 
     Relations:=[];
 
      
 
      
     // add the relation a^{n} = b^2
+
     // Add the relation a^{n} = b^2
 
     Append(Relations, [[a^(MEMORY.N)], [-b,b]]);
 
     Append(Relations, [[a^(MEMORY.N)], [-b,b]]);
 
      
 
      
     // add the relation a^{2n} = 1
+
     // Add the relation a^{2n} = 1
 
     Append(Relations, [[a^(2*MEMORY.N)], [-1]]);
 
     Append(Relations, [[a^(2*MEMORY.N)], [-1]]);
 
    
 
    
     // add the relation b^{-1}ab = a^{-1}
+
     // Add the relation b^{-1}ab = a^{-1}
 
     Append(Relations, [[b^(3),a,b],[a^(2*MEMORY.N-1)]]);
 
     Append(Relations, [[b^(3),a,b],[a^(2*MEMORY.N-1)]]);
 +
 
 
     Return Relations;
 
     Return Relations;
 
   EndDefine;
 
   EndDefine;

Revision as of 07:38, 23 August 2013

Description

The dicyclic groups are non-abelian groups with order 4n. For n = 2 the dicyclic group is isomporphic to the quarternion group Q. Note that every element of this groups can be written uniquely as a^k x^j for 0 < k < 2n and j = 0 or 1.

 Dic(n) = <a,b | a^{2n} = 1, a^{n} = b^{2}, b^{-1}ab = a^{-1}>

Reference

Coxeter, H. S. M., "7.1 The Cyclic and Dicyclic groups", Regular Complex Polytopes, Cambridge University, 1974.

Computation

 /*Use the ApCoCoA package ncpoly.*/
 
 // Number of Dicyclic group (note that  the order is 4N)
 MEMORY.N:=5;
 
 Use ZZ/(2)[a,b];
 NC.SetOrdering("LLEX");
 
 Define CreateRelationsDicyclic()
   Relations:=[];
   
   // Add the relation a^{n} = b^2
   Append(Relations, [[a^(MEMORY.N)], [-b,b]]);
   
   // Add the relation a^{2n} = 1
   Append(Relations, [[a^(2*MEMORY.N)], [-1]]);
  
   // Add the relation b^{-1}ab = a^{-1}
   Append(Relations, [[b^(3),a,b],[a^(2*MEMORY.N-1)]]);
 
   Return Relations;
 EndDefine;