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

[Stefan Kohl]

Dear Frederic,

> 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?

Yes. -- You can do the following:

gap> h:=PSL(2,5);
Group([ (3,5)(4,6), (1,2,5)(3,4,6) ])
gap> g:=AutomorphismGroup(h);
<group of size 120 with 3 generators>
gap> g_regular := Action(g,AsList(g),OnRight);
<permutation group with 3 generators>
gap> DegreeAction(g_regular);
120

and

gap> gf:=GF(64);
GF(2^6)
gap> g:=GaloisGroup(gf);
<group with 1 generators>
gap> gf_regular := Action(g,AsList(gf),OnPoints);
Group([ (3,4)(5,6,8)(7,10,9)(11,12,14,18,25,38)(13,16,21,31,51,39)
(15,19,28,45,26,40)(17,23,34,58,52,41)(20,29,48,32,53,42)(22,33,55,46,27,43)
(24,36,62,59,54,44)(30,49,35,60,56,47)(37,64,63,61,57,50) ])
gap> DegreeAction(gf_regular); # -> |GF(2^6)| - |GF(2)| = 64 - 2, prime field is fixed
62

Hope this helps,

    Stefan

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