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

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4 A sample computation with Circle

### 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

Then we compute the adjoint group of `R`:

    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

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