Representation Theory/Set representations

Alternatively, set representations are also called group actions, and we say that $G$ acts on a set $S$. Whenever $g\in G$, we will denote the corresponding element of $\mathrm{Aut}(S)$ (which are just the permutations of $S$) by $g$ as well, so that $g$ becomes a bijective function on $S$. In particular, for $x\in S$, we can make sense of expressions such as $gx$ (which shall be a shorthand for $g(x)$).

Equivalently, we could have required that for all $x,y\in S$, there exists $g\in G$ such that $gx=y$.