transitively action of stabilizer of G

Question

if $G$ acts transitively on $X$ and for a special $x$ in set $X$ we have the stabilizer of $x$ acts transitively on $X-\{x\}$ can we conclude that this proposition is true for all element of $X$?

Answer 1

Let $H := Stab(x)$, $y \in X$ and let $K := Stab(y)$. Then by transitivity, $\exists g\in G$ such that $$ g^{-1}Hg = K \text{ where } y = g^{-1}\cdot x $$ For any $z_1, z_2 \in X\setminus\{y\}$, note that $$ g\cdot z_i \in X\setminus\{x\} $$ Hence, $\exists h\in H$ such that $$ hg\cdot z_1 = g\cdot z_2 $$ So $k := g^{-1}hg \in K$ is such that $k\cdot z_1 = z_2$. Hence $Stab(y)$ acts transitively on $X\setminus\{y\}$