# Difference between revisions of "ApCoCoA-1:Quaternion group"

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− | === <div id="Quaternion_group">[[:ApCoCoA:Symbolic data#Quaternion_group|Quaternion | + | === <div id="Quaternion_group">[[:ApCoCoA:Symbolic data#Quaternion_group|Quaternion Group]]</div> === |

==== Description ==== | ==== Description ==== | ||

This particular group is non-abelian with the order 8. The Quaternion group Q has a special property: it is | This particular group is non-abelian with the order 8. The Quaternion group Q has a special property: it is |

## Latest revision as of 21:01, 22 April 2014

#### 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>