@ACuriousMind
If $S_n$ act on $M$ which is the set of all subsets of size $k$ of the set $\{1,\dots ,n\}$, then how can injectivity be ensured for all elements of $M$ as e.g. for one of the elements of $M$ say e.g. $\{1,2,3,5\}$ and act by an element in $S_n$ that is e.g. $\sigma_8=(4 6)$, then we got the mapping to $\{1,2,3,5\}$ but the identity element in $S_n$ can map $\{1,2,3,5\}$ also to itself?