sufficient conditions for faithful group action with multiple orbits

Question

Suppose group $G$ acts on a set $X$ and has multiple orbits $X_1,\ldots, X_O \subset X$. We know that $G$-action on individual orbits is isomorphic to its action on left-cosets of subgroups $H_1,\ldots, H_O \subset G$. What choice of subgroup $H_1,\ldots, H_O$ makes $G$-action faithful?

For a single orbit $G$-action is faithful when $\bigcap_{g \in G} gHg^{-1} = \{e\}$. Can we get something analogous for multiple orbits?

Answer 1

Hint: The stabilizer of a point of $X$ is of the form $gH_ig^{-1}$ for certain $g$ and $i$. The action is faithful iff every non-unit element of $G$ does not belong to some stabilizer.

Answer:

! First you need to prove that the action of $G$ is faithful if and only if the intersection of all stabilizers is the trivial subgroup $\left\{1\right\}$. This should be only a matter of looking at the definitions and rewriting them (e.g. if the action is faithful and $g$ belongs to all stabilizers, then $g\cdot x=x$ for all $x$, so $g$ acts as the identify and thus $g=1$. This is one direction.)

Then verify that any stabilizer of $G$ has the form $gH_og^{-1}$ for some $g\in G$ and some $o$. Putting these two together, the action of $G$ is faithful if and only if $\bigcap_{g\in G}\bigcap_o gH_og{-1}=\left\{1\right\}$.