ApCoCoA-1:Quaternion group

Quaternion Group

Description

This particular group is non-abelian with the order 8. The Quaternion group Q has a special property: it is Hamiltonian, that means that every subgroup of Q is a normal subgroup. An efficient representation with generators and relations is given by:

 Q_8 = <a,b | a^4 = 1, a^2 = b^2, b^{-1}ab = a^{-1}>

Reference

P.R. Girard (1984), "The quaternion group and modern physics", European Journal of Physics 5:25–32.

Coxeter, H. S. M. and Moser, W. O. J. (1980), Generators and Relations for Discrete Groups. New York: Springer-Verlag

Computation

 /*Use the ApCoCoA package ncpoly.*/
 
 Use ZZ/(2)[x,y];
 NC.SetOrdering("LLEX");

 Define CreateRelationsQuaternion()
   Relations:=[];
   // add the relation  x^4 = 1
   Append(Relations,[[x^4],[1]]);
   
   // add the relation x^2 = y^2
   Append(Relations,[[x^2],[y^2]]);
   
   // add the relation y^{-1}xy = x^{-1}
   Append(Relations,[[y^3,x,y],[x^3]]);

 Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsQuaternion();
 Gb:=NC.GB(Relations);	
 Gb;

Example in Symbolic Data Format

 <FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier">
 	<vars>x,y</vars>
 	<basis>
 	<ncpoly>x*x*x*x-1</ncpoly>
 	<ncpoly>x*x-y*y</ncpoly>
 	<ncpoly>y^3*x*y-x^3</ncpoly>
 	</basis>
 	<Comment>quaternion_group</Comment>
 </FREEALGEBRA>