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<div class="ChapSects"><a href="chap4.html#X803553D8849F6D7A">4 <span class="Heading">A sample computation with <strong class="pkg">Circle</strong></span></a>
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<h3>4 <span class="Heading">A sample computation with <strong class="pkg">Circle</strong></span></h3>
<p>Here we give an example to give the reader an idea what <strong class="pkg">Circle</strong> is able to compute.</p>
<p>It was proved in <a href="chapBib.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 <strong class="pkg">Circle</strong>. First we create two matrices that determine its generators:</p>
<table class="example">
<tr><td><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></td></tr></table>
<p>Now we construct this algebra in characteristic five and check its basic properties:</p>
<table class="example">
<tr><td><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></td></tr></table>
<p>Then we compute the adjoint group of <code class="code">R</code>. During the computation a warning will be displayed. It is caused by the method for <code class="code">IsGeneratorsOfMagmaWithInverses</code> defined in the file <code class="file">gap4r4/lib/grp.gi</code> from the <strong class="pkg">GAP</strong> library, and may be safely ignored.</p>
<table class="example">
<tr><td><pre>
gap> G := AdjointGroup( R );
#I default `IsGeneratorsOfMagmaWithInverses' method returns `true' for
[ CircleObject( [ [ 0*Z(5), Z(5), Z(5), Z(5)^3, Z(5), 0*Z(5), Z(5)^2 ],
[ 0*Z(5), 0*Z(5), 0*Z(5), Z(5), Z(5)^3, Z(5)^3, Z(5)^3 ],
[ 0*Z(5), 0*Z(5), 0*Z(5), Z(5), Z(5), 0*Z(5), Z(5) ],
[ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), Z(5), Z(5) ],
[ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), Z(5), Z(5)^3 ],
[ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ],
[ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ] ] ) ]
<group of size 15625 with 3 generators>
</pre></td></tr></table>
<p>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:</p>
<table class="example">
<tr><td><pre>
gap> f := IsomorphismPcGroup( G );;
gap> H := Image( f );
Group([ f1, f2, f3, f4, f5, f6 ])
gap> gens := MinimalGeneratingSet( H );
[ f1, f2, f5 ]
gap> gens:=List( gens, x -> UnderlyingRingElement(PreImage(f,x)));;
gap> Perform(gens,Display);
. 3 3 4 4 . 1
. . . 3 2 1 4
. . . 3 3 2 4
. . . . . 3 3
. . . . . 3 2
. . . . . . .
. . . . . . .
. 3 1 1 . . .
. . . 3 4 . 1
. . . 1 3 2 .
. . . . . 1 3
. . . . . 3 4
. . . . . . .
. . . . . . .
. 2 2 3 2 . 4
. . . 2 3 3 3
. . . 2 2 . 2
. . . . . 2 2
. . . . . 2 3
. . . . . . .
. . . . . . .
</pre></td></tr></table>
<p>It appears that the adjoint group of the algebra from example will be 3-generated in characteristic three as well:</p>
<table class="example">
<tr><td><pre>
gap> R := Algebra( GF(3), One(GF(3))*[x,y] );
<algebra over GF(3), with 2 generators>
gap> G := AdjointGroup( R );
#I default `IsGeneratorsOfMagmaWithInverses' method returns `true' for
[ CircleObject( [ [ 0*Z(3), 0*Z(3), Z(3)^0, Z(3)^0, Z(3), Z(3), 0*Z(3) ],
[ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3), Z(3)^0, Z(3)^0 ],
[ 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3), Z(3) ],
[ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ],
[ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3) ],
[ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ],
[ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ] ] ) ]
<group of size 729 with 3 generators>
gap> H := Image( IsomorphismPcGroup( G ) );
Group([ f1, f2, f3, f4, f5, f6 ])
gap> MinimalGeneratingSet( H );
[ f1, f2, f4 ]
</pre></td></tr></table>
<p>But this is not the case in characteristic two, where the adjoint group is 4-generated:</p>
<table class="example">
<tr><td><pre>
gap> R := Algebra( GF(2), One(GF(2))*[x,y] );
<algebra over GF(2), with 2 generators>
gap> G := AdjointGroup( R );
#I default `IsGeneratorsOfMagmaWithInverses' method returns `true' for
[ CircleObject( [ [ 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0 ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0 ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ] ) ]
<group of size 64 with 4 generators>
gap> H := Image( IsomorphismPcGroup( G ) );
Group([ f1, f2, f3, f4, f5, f6 ])
gap> MinimalGeneratingSet( H );
[ f1, f2, f4, f5 ]
</pre></td></tr></table>
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