Conjugation on subgroups of $A_4$ faithful?

Question

Let $X$ be the set of all subgroups of $G=A_4$. We define the group action $$G\times X\ni(g,H)\mapsto gHg^{-1}\in X$$ I am trying to determine whether this action is faithful, i.e. $\bigcap_{H\in X} Stab(H)=\{e\}$.

The subgroup $\langle(123)\rangle$ has stabiliser $\langle(123)\rangle$ and $\langle(124)\rangle$ has stabiliser $\langle(124)\rangle$. From $$\bigcap_{H\in X} Stab(H)\subseteq \langle(123)\rangle\cap \langle(124)\rangle = \{e\}$$ I deduce that the group action is faithful.

Does this look okay?

Answer 1

The easiest way is to take an $H\in X$ for which the stabilizer is trivial, for example $\{(123),(234)\}$. But your proof is also true, so of course it works just as well.