[GAP Forum] replacing group action by permutations of [1..n]

[hulpke at me.com]

Dear Frederic, Dear Forum,

> in the following two examples, GAP constructs a group of permutations, not represented as elements of the symmetric group S_n:
> 
> h:=PSL(2,5);
> g:=AutomorphismGroup(h);
> Elements(g);
> 
> 
> gf:=GF(64);
> g:=GaloisGroup(gf);
> Elements(g);
> 
> However, I would like to have an equivalent representation on respectively [1..60] and [1..64].
> 
> Is this possible?

You want the actions of the groups on the respective domains. If you ever want to get back to homomorphisms you likely will want to do this using action homomorphisms:


gap> e:=Elements(g);
[ IdentityMapping( GF(2^6) ), FrobeniusAutomorphism( GF(2^6) )^4, 
  FrobeniusAutomorphism( GF(2^6) )^2, FrobeniusAutomorphism( GF(2^6) )^3, 
  FrobeniusAutomorphism( GF(2^6) ), FrobeniusAutomorphism( GF(2^6) )^5 ]
gap> hom:=ActionHomomorphism(g,e,OnRight,"surjective");
<action epimorphism>
# the "surjective" is not strictly necessary but if not given the map will be into S6 with same image.

gap> Image(hom);
Group([ (1,5,3,4,2,6) ])

Points are numbered as the elements list `e' is.

Best,

    Alexander