WASS: Lecture 1.

Enrico ROGORA

Version 0.2, 21th August 2003

Consider a cube and enumerate its vertices {1,2,3,4,5,6,7,8}

(Front face, clockwise, starting upper left, 1,2,3,4;

Back face, clockwise, starting upper left, {5,6,7,8} .

We consider the set of orthogonal tranformations about its center.

To each of them we can associate a permutation of the set of vertices.

For example

rx: a 90 degree clockwise rotation about

the axes from the front face to the back face through its center

can be described by the permutation (1,2,3,4)(5,6,7,8)

ry: a 90 degree clockwise rotation about

the axes from the left face to the right face through its center

can be described by the permutation (1,4,8,5)(2,3,7,6)

rz: a 90 degree clockwise rotation about

the axes from the upper face to the lower face through its center

can be described by the permutation (1,5,6,2)(4,8,7,3)

Define the three permutations with GAP

gap> rx:=(1,2,3,4)(5,6,7,8);

gap> ry:=(1,4,8,5)(2,3,7,6);

gap> rz:=(1,5,6,2)(4,8,7,3);

We first consider the groups generated by the three rotations

gap> Gx:=Group(rx);

gap> Gy:=Group(ry);

gap> Gz:=Group(rz);

Of course these are three cyclic groups

gap> IsCyclic(Gx);

Hence abelian

gap> IsAbelian(Gx);

How many elements does Gx have?

gap> Size(Gx);

This is the same as the order of its generator

gap> Order(rx);

We can take the union of these three groups and the group generated by

rx, ry, rz.

gap> GG:=Union(Gx,Gy,Gz);

gap> G:=Group(rx,ry,rz);

G is strictly bigger than GG. In fact the latter

contains only element which are power of generators

but not elements like rx*ry as we can check with the

following commands

gap> rx*ry in GG;

gap> rx*ry in GG;

We now address ourselves the following problem.

Given an element of G how can we explicitely express it as a word in

the generators of G?

First, to look at all elements of G, we can use

gap> ele:=Elements(G);

This returns a list. To extract its eighth, say,

just do

gap> e:=ele[8];

If we want to express this element, say, as a word in the given generators

we can first build the free group on three letters

gap> F:=FreeGroup("x","y","z");

Then we define the homomorphism which sends the generators of F

into the generators of G

gap> hom:=GroupHomomorphismByImagesNC(F,G,GeneratorsOfGroup(F),GeneratorsOfGroup(G));

and find an element of F which is sent to it by the

morphism hom

gap> q:=PreImagesRepresentative(hom, e);

The last step is simply to translate

x into rx, y into ry and z into rz in

this last element q and we get a representation (of course not unique)

of e as aword in the generators. For example, if

q=x*y, then p=rx*ry.

gap> (1,5)(2,6)(3,7)(4,8) in G;

gap> Size(Group(rx,ry,rz,(1,5)(2,6)(3,7)(4,8)));

It is not hard to prove that G exausts all the simmetries of the cube which

are rotations. There are also reflections. If we want add them all it is sufficient

to add one to the generators, for example (1,5)(2,6)(3,7)(4,8)

What we have done so far is to "represent" the group

of rotation of the cube as a group of permutations, i.e. a

subgroup of the symmetric group with eight elements.

It is possible to choose a different representation of

this same group as a group of simmetries in less elements.

It is enough, for example to consider the action induced over

the 6 faces of the cube, which we can enumerate like this.

Front 1

Back 6

Right 3

Left 4

Bottom 5

Upper 2

In terms of this new representation we get,

gap> Rx:=(2,3,5,4);

gap> Ry:=(1,2,6,5);

gap> Rz:=(1,3,6,4);

The group generated by these element is now a subgroup of the group

of simmetries over six elements.

gap> G2:=Group(Rx,Ry,Rz);

For checking with gap that the two groups are indeed isomorphic

we first build an homomorphism betwwen them that sends the generators

rx, ry, rz into Rx, Ry, Rz

and then we check that is isomorphism is injective and surjective.

This is done with the following commanda

gap> hom:=GroupHomomorphismByImagesNC(G,G2,GeneratorsOfGroup(G),GeneratorsOfGroup(G2));

gap> Kernel(hom);

gap> Image(hom);

The Kernel is only the identity and the image coincides with G2