TITLE: injectivity of a group action
QUESTION [5 upvotes]: The action of a group $G$ on $X$ is always "injective" in the following sense:
if $x\not = y$ then $\forall g\in G$, $gx\not = gy$ indeed if $gx=gy$ then $g^{-1}(gx)=(g^{-1})gx=x=g^{-1}(gy)=(g^{-1}g)y=y$. Is this why the second axiom of group action is set: $g(hx)=(gh)x$? and what is the importance of this "injectivity"?

REPLY [5 votes]: A $G$-action on $X$ is the same as a group morphism $G\to S_X$ into the symmetric group on $X$. Thus each element $g\in G$ induces a permutation $x\mapsto gx$ of $X$, and group multiplication corresponds to permutation composition. In particular, $x\mapsto gx$ being a bijection (=permutation), with inverse $y\mapsto g^{-1}y$, means it is both surjective and injective.
I guess my 'point' is that you shouldn't only concentrate on the injectivity, but rather think of permutations of the set.