When are two elements of a group equivalent in terms of their action on an element of an orbit of X?

Question

I have strong intution that if $g_1$ and $g_2$ are two members of a group $G$ acting on $X$; then, if $g_1(x)= g_2(x)$ for some $x$ in a $G$-orbit, $O$ of $X$; then $g_1(x)=g_2(x)$ for all $x$ in $O$. But I can't prove this.

I am using this as a preliminary result to derive that each orbit corresponds to a subgroup of G, so only want to use the definition of orbits and actions to prove this result.

Any help is appreciated.

Answer 1

Let $G=S_3$ act on $X=O=\{1,2,3\}$ Both $(23)$ and $e$ fix $1$ but do different things to the other elements of $O$.

Answer 2

Note that, by action properties:

\begin{alignat}{1} &g_1\cdot x=g_2\cdot x \iff \\ &(g_2^{-1}g_1)\cdot x=x \iff \\ &g_2^{-1}g_1\in G_x \\ \end{alignat}

where $G_x$ is the stabilizer of $x\in X$. Since the stabilizers of the points of one same orbit are conjugate, a sufficient condition for your claim to hold is that $G_x\unlhd G$, as in this case $G_{x'}=G_x$ for every $x'\in O(x)$.