Group_Action_on_Sets_with_k_Elements

Let $\struct {G, \circ}$ be a finite group whose identity is $e$. Let \Bbb S = \set {S \subseteq G: \card S = k}, that is, the set of all of subsets of $G$ which have exactly $k$ elements. Let $G$ act on $\Bbb S$ by the rule $\forall S \in \Bbb S: g * S = g S = \set {x \in G: x = g s: s \in S}$ This is a group action, and $\forall S \in \Bbb S: \order {\Stab S} \divides \card S$ where $\Stab S$ denotes the stabilizer of $S$ by $G$.