Consider the action of $\operatorname{Aut}(\Gamma)$ of a graph $\Gamma$ on the edges of $\Gamma$. My book states that this is not always a permutation group.
Question 1: Why is this not always a permutation group?
Now the following definition is given
Consider a group $G$ and a set $X$. $(G,X)$ is called a permutation representation if
each element $g \in G$ defines a permutation of $X$ (the image under that permutation of an element $x\in X$ is written as $x^g$) and
the operation (mostly composition) in $G$ corresponds with the composition as permutations, i.o.w., if $\forall g,h\in G,x\in X: x^{(gh)}=(x^g)^h$.
Question 2: Are the elements in $G$ permutations themselves or are they mapped to a permutation? (This is a very confusing definition, so maybe somebody could help me out with an alternative one, please).
Question 3: What is the difference between a permutation group and a permutation representation. They seem to both contain permutations (well, if the answer to question 2 is that $G$ contains permutations).
Thanks.