[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