Let $G$ be a group and let $S=G.$ Recall that $g\cdot x∶=gx$ for $g \in G$ and $x \in S$ is an action of $G$ on $S.$
We defined the function already in class fyi.
We were also given this bit of information : Recall that $Sym(S)$ = {σ ∶ σ∶ S → S is a bijection} is a group under composition and that the identity element in $Sym(G)$ is the identity function $idS$ that sends $x$ to $x$ and what is the value of $fg(eG)?$
The only thing I can think of to use is if you have function that is a homomorphism of groups $G \rightarrow H$ then you can show it is injective by showing if $f(a)=e_H$ this implies that $a=e_G$.
Best Answer
$S=G$
$(g,x) \mapsto gx$ is action of $G$ on $S$,
For fixed $g\in G$, $g$ permutes elements of $S$ by above action which can be identify as element $\pi_g \in \text{Sym}(S)$.
This induces homomorphism $f: G \to \text{Sym}(S)$ defined as $g \mapsto \pi_g$ with $K=\text{Ker}(f)$.
Now, $g \in K$ $iff$ $g$ fixes every element of $S$ by above action.
And the only such element is $e \in G$.