Maximal subgroups of a finite group Let $G$ be a finite group acting on a finite set $X$. If the action is primitive  then the stabilizers are maximal subgroups of $G$ (converse also true).
Is there any criteria to get maximal subgroups as stabilizers, with some restriction on the "action"?
For ex. the action of $S_3$ on $X=\{1,2,3\}$ is primitive, and so stabilizers are maximal subgroups of $G$, which are Sylow-2 subgroups of $S_3$. But how can we obtain the maximal subgroup $\langle (123)\rangle$?
 A: To get an action of $S_3$ in which $\langle(1,2,3)\rangle$ is the stabilizer of a point, take $X=\{1,-1\}$, and let $S_3$ act by parity: even permutations fix both $1$ and $-1$, odd permutations swap $1$ and $-1$. Then the stabilizer of a single point, $1$, is $A_3 = \langle (1,2,3)\rangle$.
In general, if $H$ is a subgroup of $G$, then $G$ acts on the left cosets of $H$ by left multiplication, $g\cdot xH = gxH$. The stabilizer of the coset $H$ is $H$ itself. If you have a partition of the cosets so that $G$ acts on the partition, let $K = \{ g\in G\mid gH\text{ is in the same block as }H\}$ be the collection of all elements that represent cosets in the same part of the partition as $H$. Then $K$ certainly contains $H$. I claim that $K$ is a subgroup of $G$: for if $x\in K$, then $xH$ is the same block of the partition as $H$, hence $x^{-1}(xH)$ is in the same block as $x^{-1}H$. But $x^{-1}(xH) = H$, so $x^{-1}H$ is also in the same block of the partition, hence $x^{-1}\in K$. That is, $K$ is closed under inverses. And if $x,y\in K$, $x^{-1}\in K$. Taking the cosets $x^{-1}H$ and $yH$, which are in the same block, and multiplying by $x$, we conclude that $xyH$ and $H$ are in the same block, so $xy\in K$. Thus, $K$ is a subgroup, $H\leq K \leq G$.
Conversely, if $K$ is a subgroup, $H\leq K\leq G$, then $K$ induces a partition of the cosets into blocks for the action of $G$: just take the cosets of $K$, and partition each into cosets of $H$.
So systems of blocks in the action of $G$ on the left cosets of $H$ correspond to subgroups $K$, $H\leq K\leq G$.
Hence, if $H$ is a maximal subgroup, then the action of $G$ on the cosets is primitive, since the only possible systems of blocks are the trivial and total blocks, corresponding to $H$ and to $G$. 
In particular, every maximal subgroup of $G$ is the stabilizer of a point in a primitive action of $G$, namely, $H$ is the stabilizer of $H$ in the (left) action of $G$ on the left cosets of $H$ in $G$. 
A: Here is an older version of Arturo's answer:
The symmetric group of degree n acts on polynomials in n variables.  We can view $\operatorname{Sym}(3)$ as acting on the variables $x,y,z$.
Consider its action on the polynomial $x+y$.  The orbit contains $x+y$, $x+z$, and $y+z$.  The stabilizer of $x+y$ is the maximal subgroup $\operatorname{Sym}(\{x,y\})\cong\operatorname{Sym}(2)$.
Consider its action on the polynomial $\Delta = (x−y)(x−z)(y−z)$.  The orbit has only two polynomials, $\Delta$ and $-\Delta$.  The stabilizer of $\Delta$ is the maximal subgroup $\operatorname{Alt}(\{x,y,z\})\cong\operatorname{Alt}(3)$.
You can play similar tricks with any subgroup of $\operatorname{Sym}(n)$.  This is one way to look at "resolvents".
