I am studying for an exam, and I got stuck on question on finding a certain homomorphism.
For a group $G$, and a finite subset $A \subset G$, with $n=\#A$, consider subgroup $H=\{ g \in G \mid \text{for all }a \in A, gag^{−1}\in A\}$, and $N=\{g\in G\mid \text{for all }a\in A, gag^{-1}=a\}$ ($N$ is a normal subgroup of $H$). Then I have to find a homomorphism $f : H \to S_n$ such that $N = \ker(f)$.
I do not really know how to start, in general I find it a bit hard grasp the concept of a function from a group to a permutation. Anyone that can help me get started on this?
The main clue here is that the subgroup $N$, which is the kernel of the map, is about conjugation, which is a standard action you will have learned about in your course. Actions are very relevant here, as an action of a group $H$ on a set $X$ is really just a map $H\rightarrow \mathrm{Sym}(X)$, and vice-versa (such homomorphisms define actions). Indeed, the map you are looking for correponds to the conjugation action, so we have the map $g\mapsto (a\mapsto g^{-1}ag, a\in A)$.
Here, $(a\mapsto g^{-1}ag)$ defines a permutation of the set $A$, and so an element of $S_n$. The kernel is your subgroup $N$ as this permutation is the identity map permutation if and only if $g\in N$.