[GAP Forum] help

[Max Horn]

Am 04.01.2011 um 06:39 schrieb Sara Radfar:

> Hi
> We know that in gap we use notation $PSL(p,q)$ for projective linear
> group.What is
> notation projective semi-linear group?
> Thanks
> sara

Dear Sara,

to the best of my knowledge, there is no function for that in GAP. (Also note that some people would consider PGL to be the project linear group; I guess it's a matter of taste or the field you work in; I only mention it for completeness).

However, you can create the required group yourself as follows. This uses that the automorphism group of a finite field is cyclic, and generated by the Frobenius automorphism.


# Parameters we are interested in
n:=3;; q:=9;; F:=GF(q);;
# To construct PSL, start with SL; to construct PGL, start with GL.
G := GL(n,F);
# Pick a normed basis vector, and take the orbit of the subspace it spans, under the action of G
e1:=IdentityMat(n,F)[1];;
orb:=Set(Orbit(G,e1,OnLines));;
# Finally construct PSL(n,F) by turning this action into a permutation representation for it.
psl:=Action(G,orb,OnLines);;

# You can now work with this, and e.g. test if the size is correct.
Size(psl) = Size(PSL(n,q));

# Now, we can turn the Frobenius automorphism into a permutation of the set 'orb'
aut:=FrobeniusAutomorphism(F);;
frob:=PermList(List(orb,v->Position(orb, List(v,x->Image(aut,x)))));;

# Finally, by adding 'frob' as a new generator to psl, we get the desired group
H:=ClosureGroupAddElm(psl, frob);


Hope that helps,
Max