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`

:

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:

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:

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:

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

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