GAP (Circle) - Chapter 3: A sample computation with Circle

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### 3. 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`. During the computation a warning will be displayed. It is caused by the method for `IsGeneratorsOfMagmaWithInverses` defined in the file `gap4r4/lib/grp.gi` from the GAP library, and may be safely ignored.

    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>

Now we can find the minimal generating set of `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 );
    [ 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
     . . . . . . .
     . . . . . . .

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

    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 ]

But this is not the case in characteristic two, 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 );
    #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 ]

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