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<h3>4 A sample computation with Circle</h3>

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

It was proved in <a href="chapBib_mj.html#biBKazarin-Soules-2004">[KS04]</a> 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:

<div class="example"><pre>

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 ] ];;

</pre></div>

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

<div class="example"><pre>

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

</pre></div>

Then we compute the adjoint group of <code class="code">R</code>:

<div class="example"><pre>

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

</pre></div>

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

<div class="example"><pre>

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

</pre></div>

One can also use <code class="code">UnderlyingRingElement(PreImage(f,x))</code> to find the preimage of <code class="code">x</code> in <code class="code">G</code>.

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

<div class="example"><pre>

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

</pre></div>

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

<div class="example"><pre>

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

</pre></div>

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