chap4.txt

  
  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:
  
    Example  
    
    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:
  
    Example  
    
    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:
  
    Example  
    
    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:
  
    Example  
    
    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:
  
    Example  
    
    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:
  
    Example  
    
    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