GAP (Circle) - Chapter 4: A sample computation with Circle

Here we give an example to give the reader an idea what Circle is able to compute.

It was proved in [KS04] that if \(R\) is a finite nilpotent two-generated algebra over a field of characteristic \(p>3\) whose adjoint group has at most three generators, then the dimension of \(R\) is not greater than 9. Also, an example of the 6-dimensional such algebra with the 3-generated adjoint group was given there. We will construct the algebra from this example and investigate it using Circle. First we create two matrices that determine its generators:


gap> x:=[ [ 0, 1, 0, 0, 0, 0, 0 ],
>         [ 0, 0, 0, 1, 0, 0, 0 ],
>         [ 0, 0, 0, 0, 1, 0, 0 ],
>         [ 0, 0, 0, 0, 0, 0, 1 ],
>         [ 0, 0, 0, 0, 0, 1, 0 ],
>         [ 0, 0, 0, 0, 0, 0, 0 ],
>         [ 0, 0, 0, 0, 0, 0, 0 ] ];;
gap> y:=[ [ 0, 0, 1, 0, 0, 0, 0 ],
>         [ 0, 0, 0, 0,-1, 0, 0 ],
>         [ 0, 0, 0, 1, 0, 1, 0 ],
>         [ 0, 0, 0, 0, 0, 1, 0 ],
>         [ 0, 0, 0, 0, 0, 0,-1 ],
>         [ 0, 0, 0, 0, 0, 0, 0 ],
>         [ 0, 0, 0, 0, 0, 0, 0 ] ];;

Now we construct this algebra in characteristic five and check its basic properties:


gap> R := Algebra( GF(5), One(GF(5))*[x,y] );
<algebra over GF(5), with 2 generators>
gap> Dimension( R );
6
gap> Size( R );
15625
gap> RadicalOfAlgebra( R ) = R;
true


gap> G := AdjointGroup( R );;
gap> Size(G);
15625

Now we can find the generating set of minimal possible order for the group G, and check that G it is 3-generated. To do this, first we need to convert it to the isomorphic PcGroup:


gap> f := IsomorphismPcGroup( G );;
gap> H := Image( f );
Group([ f1, f2, f3, f4, f5, f6 ])
gap> gens := MinimalGeneratingSet( H );;
gap> Length( gens );
3

One can also use UnderlyingRingElement(PreImage(f,x)) to find the preimage of x in G.

It appears that the adjoint group of the algebra from example will be 3-generated in characteristic 3 as well:


gap> R := Algebra( GF(3), One(GF(3))*[x,y] );
<algebra over GF(3), with 2 generators>
gap> G := AdjointGroup( R );;
gap> H := Image( IsomorphismPcGroup( G ) );
Group([ f1, f2, f3, f4, f5, f6 ])
gap> Length( MinimalGeneratingSet( H ) );
3

But this is not the case in characteristic 2, where the adjoint group is 4-generated:


gap> R := Algebra( GF(2), One(GF(2))*[x,y] );
<algebra over GF(2), with 2 generators>
gap> G := AdjointGroup( R );;
gap> Size(G);
64
gap> H := Image( IsomorphismPcGroup( G ) );
Group([ f1, f2, f3, f4, f5, f6 ])
gap> Length( MinimalGeneratingSet( H ) );
4

4 A sample computation with Circle