Let a group $G$ act on a space $S$. If a point $p$ in the space remains invariant under the action of a subgroup of $G$, we call the subgroup the stabilizer of $p$.
$\text{Stab}(p) = \{g: g \in G, \, p\in S, \, g.p = p\}$
Can the notion of stabilizer be extended to a set of points (say $P$) in the space rather than a single point? i.e. can we define a subgroup in the following way?
$\{g: g \in G, \, P\subset S, \, g.p_i = p_j \, \forall \, p_i, p_j \in P\}$
If so, what is such a group called?
For a subset $S'$ of $S$, define $Stab_1(S') = \bigcap_{p\in S'} Stab(p)$ which is the point-wise stabilizer of $S'$. This is a subgroup of $G$ since the intersection of subgroups is itself a subgroup.
Another set to consider is $Stab_2(S') = \{g\in G\mid g\cdot S'\subseteq S'\}$ which stabilizes the subset $S'$. Can you check whether this set is a subgroup of $G$?