Let $\sigma:G\rightarrow S_X$ be a transitive group action with $|X|\geq 2$. If $x\in X$, denote $G_x$ as the stabiliser of $x$ in $G$.
Show that if $G_x$ is transitive on $X\setminus\{x\}$ for some $x\in X$, then $G_x$ is transitive on $X\setminus\{x\}$ for every $x\in X$.
I tried to show it by means of contraposition but it didn't workout. Help will really be appreciated. Thanks in advance!!
OK, let $y\in X$, and $w_1,w_2\in X\setminus \{y\}$. We need to find a $t\in G$ which carries $y$ to $y$ and $w_1$ to $w_2$.
Choose $h\in G$ that carries $y$ to the special $x$. Now we have $w_1^{h}, w_2^{h}\in X\setminus \{x\}$; so there is an element $k\in G_x$ which carries $w_1^{h}$ to $w_2^{h}$.
The element $t=hkh^{-1}$ now does what we want.
Think "similar matrices and change of basis", "conjugate permutations have same shape".
To simplify notation, I will write $g \cdot x$ for the image of $x \in X$ under $\sigma(g)$ with $g \in G$.
Suppose that $G_x$ is transitive on $X \setminus \{x\}$, and let $y \in X$. By transitivity there exists $g \in G$ such that $g \cdot x = y$. Now let $u,v \in Y \setminus \{ y \}$. Then $g^{-1}\cdot u,\, g^{-1} \cdot v \in X \setminus \{ x \}$, so there exists $h \in G_x$ such that $hg^{-1} \cdot u = g^{-1} \cdot v$. Hence $ghg^{-1} \cdot u = v$ and $ghg^{-1} \in G_y$, so we have shown that $G_y$ is transitive on $Y \setminus \{y\}$.