  
  [1X4 A sample computation with [5XCircle[1X[0X
  
  Here  we  give  an example to give the reader an idea what [5XCircle[0m 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 [5XCircle[0m. First we create two matrices that determine its
  generators:
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X                                                                                          [0X
    [4Xgap> x:=[ [ 0, 1, 0, 0, 0, 0, 0 ],[0X
    [4X>         [ 0, 0, 0, 1, 0, 0, 0 ],[0X
    [4X>         [ 0, 0, 0, 0, 1, 0, 0 ],[0X
    [4X>         [ 0, 0, 0, 0, 0, 0, 1 ],[0X
    [4X>         [ 0, 0, 0, 0, 0, 1, 0 ],[0X
    [4X>         [ 0, 0, 0, 0, 0, 0, 0 ],[0X
    [4X>         [ 0, 0, 0, 0, 0, 0, 0 ] ];;[0X
    [4Xgap> y:=[ [ 0, 0, 1, 0, 0, 0, 0 ],[0X
    [4X>         [ 0, 0, 0, 0,-1, 0, 0 ],[0X
    [4X>         [ 0, 0, 0, 1, 0, 1, 0 ],[0X
    [4X>         [ 0, 0, 0, 0, 0, 1, 0 ],[0X
    [4X>         [ 0, 0, 0, 0, 0, 0,-1 ],[0X
    [4X>         [ 0, 0, 0, 0, 0, 0, 0 ],[0X
    [4X>         [ 0, 0, 0, 0, 0, 0, 0 ] ];;[0X
    [4X                                                                                                [0X
  [4X------------------------------------------------------------------[0X
  
  Now  we  construct  this  algebra in characteristic five and check its basic
  properties:
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X                                                                                          [0X
    [4Xgap> R := Algebra( GF(5), One(GF(5))*[x,y] );[0X
    [4X<algebra over GF(5), with 2 generators>[0X
    [4Xgap> Dimension( R );[0X
    [4X6[0X
    [4Xgap> Size( R );[0X
    [4X15625[0X
    [4Xgap> RadicalOfAlgebra( R ) = R;[0X
    [4Xtrue[0X
    [4X                                                                                                [0X
  [4X------------------------------------------------------------------[0X
  
  Then  we  compute  the  adjoint group of [10XR[0m. During the computation a warning
  will    be    displayed.    It    is    caused    by    the    method    for
  [10XIsGeneratorsOfMagmaWithInverses[0m  defined  in the file [11Xgap4r4/lib/grp.gi[0m from
  the [5XGAP[0m library, and may be safely ignored.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X [0X
    [4Xgap> G := AdjointGroup( R );[0X
    [4X#I  default `IsGeneratorsOfMagmaWithInverses' method returns `true' for [0X
    [4X[ CircleObject( [ [ 0*Z(5), Z(5), Z(5), Z(5)^3, Z(5), 0*Z(5), Z(5)^2 ],[0X
    [4X      [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5), Z(5)^3, Z(5)^3, Z(5)^3 ],[0X
    [4X      [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5), Z(5), 0*Z(5), Z(5) ],[0X
    [4X      [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), Z(5), Z(5) ],[0X
    [4X      [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), Z(5), Z(5)^3 ],[0X
    [4X      [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ],[0X
    [4X      [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ] ] ) ][0X
    [4X<group of size 15625 with 3 generators>[0X
    [4X                                                                                                [0X
  [4X------------------------------------------------------------------[0X
  
  Now  we  can find the generating set of minimal possible order for the group
  [10XG[0m,  and check that [10XG[0m it is 3-generated. To do this, first we need to convert
  it to the isomorphic PcGroup:
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X [0X
    [4Xgap> f := IsomorphismPcGroup( G );;[0X
    [4Xgap> H := Image( f );[0X
    [4XGroup([ f1, f2, f3, f4, f5, f6 ])[0X
    [4Xgap> gens := MinimalGeneratingSet( H );[0X
    [4X[ f1, f2, f5 ][0X
    [4Xgap> gens:=List( gens, x -> UnderlyingRingElement(PreImage(f,x)));;[0X
    [4Xgap> Perform(gens,Display);                                        [0X
    [4X . 3 3 4 4 . 1[0X
    [4X . . . 3 2 1 4[0X
    [4X . . . 3 3 2 4[0X
    [4X . . . . . 3 3[0X
    [4X . . . . . 3 2[0X
    [4X . . . . . . .[0X
    [4X . . . . . . .[0X
    [4X . 3 1 1 . . .[0X
    [4X . . . 3 4 . 1[0X
    [4X . . . 1 3 2 .[0X
    [4X . . . . . 1 3[0X
    [4X . . . . . 3 4[0X
    [4X . . . . . . .[0X
    [4X . . . . . . .[0X
    [4X . 2 2 3 2 . 4[0X
    [4X . . . 2 3 3 3[0X
    [4X . . . 2 2 . 2[0X
    [4X . . . . . 2 2[0X
    [4X . . . . . 2 3[0X
    [4X . . . . . . .[0X
    [4X . . . . . . .[0X
    [4X                                                                                                [0X
  [4X------------------------------------------------------------------[0X
  
  It  appears  that  the  adjoint  group  of  the algebra from example will be
  3-generated in characteristic three as well:
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> R := Algebra( GF(3), One(GF(3))*[x,y] );[0X
    [4X<algebra over GF(3), with 2 generators>[0X
    [4Xgap> G := AdjointGroup( R );[0X
    [4X#I  default `IsGeneratorsOfMagmaWithInverses' method returns `true' for [0X
    [4X[ CircleObject( [ [ 0*Z(3), 0*Z(3), Z(3)^0, Z(3)^0, Z(3), Z(3), 0*Z(3) ],[0X
    [4X      [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3), Z(3)^0, Z(3)^0 ],[0X
    [4X      [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3), Z(3) ],[0X
    [4X      [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ],[0X
    [4X      [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3) ],[0X
    [4X      [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ],[0X
    [4X      [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ] ] ) ][0X
    [4X<group of size 729 with 3 generators>[0X
    [4Xgap> H := Image( IsomorphismPcGroup( G ) );[0X
    [4XGroup([ f1, f2, f3, f4, f5, f6 ])[0X
    [4Xgap> MinimalGeneratingSet( H );[0X
    [4X[ f1, f2, f4 ][0X
    [4X                                                                                                [0X
  [4X------------------------------------------------------------------[0X
  
  But  this  is not the case in characteristic two, where the adjoint group is
  4-generated:
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> R := Algebra( GF(2), One(GF(2))*[x,y] );[0X
    [4X<algebra over GF(2), with 2 generators>[0X
    [4Xgap> G := AdjointGroup( R );                   [0X
    [4X#I  default `IsGeneratorsOfMagmaWithInverses' method returns `true' for [0X
    [4X[ CircleObject( [ [ 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],[0X
    [4X      [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2) ],[0X
    [4X      [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ],[0X
    [4X      [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0 ],[0X
    [4X      [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0 ],[0X
    [4X      [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],[0X
    [4X      [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ] ) ][0X
    [4X<group of size 64 with 4 generators>[0X
    [4Xgap> H := Image( IsomorphismPcGroup( G ) );[0X
    [4XGroup([ f1, f2, f3, f4, f5, f6 ])[0X
    [4Xgap> MinimalGeneratingSet( H );[0X
    [4X[ f1, f2, f4, f5 ][0X
    [4X                                                                                                [0X
  [4X------------------------------------------------------------------[0X
  
