Computing the Stabilizer of a coset $Nh$ in $H=N.G$ using GAP.

Question

Let $H=N.G$ be an extension of $N$ by $G$ and $Nh$, $h\in H$, a coset of $N$ in $H$. How do one compute the stabilizer $S$ of $Nh$ in $H$ using GAP?

Answer 1

What you want to calculate is the full pre-image of the centralizer in the factor group. You can either go through a natural homomorphism:

gap> H:=SymmetricGroup(4);;
gap> N:=Core(H,SylowSubgroup(H,2));;h:=(1,2);; # E.g. V4 in S4
gap> nat:=NaturalHomomorphismByNormalSubgroup(H,N);
[ (1,2,3,4), (1,2) ] -> [ f1*f2, f1 ]
gap> cen:=Centralizer(Image(nat),Image(nat,h));
Group([ f1 ])
gap> s:=PreImage(nat,cen);
Group([ (3,4), (1,2)(3,4), (1,3)(2,4) ])
gap> Size(s);
8

or use the dedicated function CentralizerModulo:

gap> s:=CentralizerModulo(H,N,h);
Group([ (3,4), (1,4)(2,3), (1,3)(2,4) ])