[GAP Forum] Work with PGammaL(n,q), APGammaL(n,q) and PSL.

[Alexander Hulpke]

Dear  GAP-Forum,

Andreas Rabenstetter asked:

> Question1: I wanted to work with:
> 
> 
> PGammaL(n,q) or APGammaL(n,q):=<PGammaL(n,q), \tau>,
> \tau is the Dynkin Automorphism, and  it´s supgroup the Socle PSL(n,q).
> 
> 
> Therefore I defined the Group APGammaL(n,q) with the command "W:=
> PSL(n,q);   G:=AutomorphismGroup(W); " . Since the PSL(n,q) (with the
> command "PSL(n,q)") isn´t regarded as a supgroup of AGammaL(n,q) or
> PGammaL(n,q), I have created the supgroup with the command "Socle( G )".
> Now my Computer doen´t calculate the Groups for (n,q) which are not even
> very high, because it is to much to calculate for the socle or the
> Autormorphism Group. Now I wanted to know, if it´s possible to create the
> whole situation with other commands which work faster. Or does GAP even
> know the structure.

at the moment GAP does not treat these groups specially but runs in the generic calculations for socle and automorphism group. As these are permutation groups of moderately large degree q^n and large order this takes a while.
I have code in the development version that immediately writes down generators of the automorphism group, it will be part of GAP 4.8.

The easiest way to do equivalently is probably to use the homomorphism SL->PSL explicitly:

gap> sl:=SL(5,7); 
SL(5,7)
gap> vecs:=NormedVectors(GF(7)^5);;
gap> hom:=ActionHomomorphism(sl,vecs,OnLines,"surjective");
<action epimorphism>
gap> psl:=Image(hom);
<permutation group with 2 generators>

now take generators of the automorphisms for sl:

# induced by GL
gap> auts:=List(GeneratorsOfGroup(gl),x->GroupHomomorphismByImagesNC(sl,sl,GeneratorsOfGroup(sl),
> List(GeneratorsOfGroup(sl),y->y^x)));;
# graph automorphism
gap> Add(auts,GroupHomomorphismByImagesNC(sl,sl,GeneratorsOfGroup(sl),                                    
> List(GeneratorsOfGroup(sl),y->TransposedMat(y)^-1)));;

If the field was nonprime, you need to add a generator of the Galois automorphisms.

Transfer to psl:
auts:=List(auts,x->InducedAutomorphism(hom,x));;

and make a group out of it:

gap> A:=Group(auts);;                                 
gap> SetIsGroupOfAutomorphismsFiniteGroup(A,true); # will trigger methods for automorphism groups

Note that a command such as `IsomorphismPermGroup' for A still will take a little while.

> Question2: It would be possible for me to work not porjective. Then I have
> defined GammaL(n,q) with the command GAP knows. But how can I get the
> subgroup X, which is isomorph to SL(n,q)? I have tried the command solce,
> but then for (n,q)=(3,2^4) I get the wrong group:
> 
> gap> G:=GammaL(3,2^4);
> GammaL(3,16)
> gap> Y:=Socle( G );
> <group of 12x12 matrices in characteristic 2>
> gap> IsSubgroup(G,Y);
> true
> gap> Size(Y);
> 15

No, that is the socle of GammaL (=its centre). You want the socle modulo the centre, probably PerfectResiduum will be easiest.

Best,

    Alexander Hulpke




-- Colorado State University, Department of Mathematics,
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