This question is related to my trouble in abstract algebra in general so some general advice is really appreciated!! (I'm really new to this.)
Let A be an infinite subset of G with $n = \#A$. \begin{equation} H = \{g∈G: gag^{-1}∈A, \forall a∈A\}\\ N = \{g∈G: ga=ag, \forall a∈A \} \end{equation}
First I'm asked to prove that H is a subgroup of G. No problem. first I use a=g to prove that H isn't empty, then I fix a and take g and g'. Second I'm asked to prove that N is a normal subgroup of H. Now I have to show that any left coset $hN=Nh$ with h in H. (my first complication)
And in the end I have to find a homomorphism $\,f:H \to S_n$ so that $N=Ker(f)$. Now with this one I have no idea how to even begin.