Let $G$ be a finite group, $S$ be the set of all subsets of $G$ of size $n$, and for $g \in G$, $T \in S$ define $g.T=\{gt: t \in T\}$.
My course's notes says that this is a group action of $G$ on $S$, and "shows" it by first stating without proof that $g.T$ is also of size $n$, hence in $S$. To me it's not immediately obvious that this is true.
For it to be true requires $gt_1 = gt_2 \implies t_1=t_2$, i.e. two distinct elements of $T$ will always be mapped to distinct values by $g$. If $g$ maps two distinct elements of $T$ to the same value then the cardinality of $g.T$ will be less than $n$.
Why is it impossible for $g$ to map two distinct values $t_1, t_2$ to the same value?
Suppose $gt_1=gt_2$. Multiply both sides by $g^{-1}$ from the left side and you will get $t_1=t_2$.