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 $$\operatorname{Aut}(S)$$ (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 $$gx = y$$.