In transformation group $($$X$,$G$$)$ where $X$ is compact and hausdorff suppose that for any $x$ $\in$ $X$ , $xG$ is finite. we can see that :
$xG$$=$ $\overline {xG}$$=$$x$$E(X,G)$
{$xE(X,G)$ $|$ $x$ $\in$ $X$ } is a partition of $X$ to it's minimal subsets
My question : Is $($$X$,$G$$)$ distal ?
Suppose $x, y \in X$ are proximal. So, $$\inf_{g \in G}\{\rho(xg,yg)\}=0.$$ Then let $g_0, g_1 \ldots$ be a sequence of elements of $G$ such that $\lim_{i \to \infty}(\rho(xg_i,yg_i))=0$. As the orbits of $x$ and $y$ are both finite, there are only finitely many possible values for the real number $\rho(xg_i,yg_i)$ and hence there exists a group element $\hat{g}$ among the $g_i$ such that $\rho (x\hat{g},y\hat{g}) = 0$. It follows that $x\hat{g} = y\hat{g}$, hence $x = y$ and so $(X,G)$ is distal.