Group Action as permutations I'm trying to study on Group actions. the paper says(if I understand) that if I have Set $S$ and an action $\alpha$: $G \times S \rightarrow S$ . 
then the action may be viewed as permutation by $x \rightarrow \alpha(g,x)$.
The thing is after reading the definition I can't see why it is not possible that $\alpha(g,x_1) = \alpha(g,x_2) $ where $x_1 \neq x_2$ , and thus we don't have a permutation.
your help is appropriated  
 A: I suppose that what your reference is saying is that an action induces for every $g \in G$ a permutation on $S$ defined by the mapping 
$$x \mapsto \alpha(g,x)\ ,$$
where $g$ is fixed.
This mapping is indeed a permutation because it has an inverse, namely the mapping defined as 
$$x \mapsto \alpha(g^{-1},x)\ .$$
If you call the first mapping $\varphi_g$ and the second one $\varphi_{g^{-1}}$ a simple computation shows that for every $x \in S$
$$\varphi_{g^{-1}}(\varphi(g))=\alpha(g^{-1},\alpha(g,x))=\alpha(g^{-1}g,x)=\alpha(1,x)=x$$
where the last equalities follow from the definition of action of a group.
The mapping that associates to every $g \in G$ the permutation $\varphi_g$ is an homomorphism from $G$ to the group $\text{Aut}(S)$, the group of permutation of $S$.
The converse also holds: for every $\varphi \colon G \to \text{Aut}(S)$ you obtain the mapping 
$$\alpha \colon G \times S \to S$$
defined by $\alpha(g,x)=\varphi_g(x)$.
With some easy computations you can prove that, since $\varphi$ is an homomorphism, $\alpha$ is indeed an action of $G$ on $S$.
This is actually a very common phenomena in mathematics: usually to every kind of algebraic structure (monoid, group, ring, algebra) is associated a notion of action which turns out to be equivalent to the notion of an homomorphism from the structure to a structure made out of endo/auto-morphisms. 
Also if you know about currying then an action can be defined as the uncurried version of an homomorphism into an endomorphism structure.
Hope this helps.
A: $\alpha(g,x_1) = \alpha(g,x_2)$ implies $x_1=x_2$, since by the definition of a group action 
$$
\alpha(g^{-1},\alpha(g,x)) = \alpha(g^{-1}g,x)=\alpha(e,x)=x\;.
$$
