Labelling of the elements of $\Theta \le S_n$ such that also $\Gamma_\Theta \le S_n$

Question

Let $\Theta=\{\theta_1,\dots,\theta_n\}$ be a subgroup of $S_n$, the symmetric group of degree $n$. Let's define $\Gamma_\Theta=\{\gamma_1,\dots,\gamma_n\}$ by:

$$\gamma_j(k):=\theta_k(j), \quad \forall j,k=1,\dots,n \tag 1$$

Lemma. $\Gamma_\Theta \subseteq S_n$ if and only if:

$$\theta_j(i)=\theta_k(i) \Rightarrow \theta_j=\theta_k \tag 2$$

Proof. If $\gamma_i \in S_n$, then $\theta_j(i)=\theta_k(i) \Rightarrow \gamma_i(j)=\gamma_i(k) \Rightarrow j=k \Rightarrow \theta_j=\theta_k$, so that $(2)$ holds. Conversely, if $(2)$ holds, then $\gamma_i(j)=\gamma_i(k) \Rightarrow \theta_j(i)=\theta_k(i) \Rightarrow \theta_j=\theta_k \Rightarrow j=k$, so that $\gamma_i$ is injective and then bijective. $\quad \Box$

Now, I suppose that there is some (unique?) labelling of the elements of $\Theta$ such that $\Gamma_\Theta \le S_n$; I've verified it directly for $n=3$ (and then necessarily $\Theta=\{(123),(231),(312)\}$ [*]), but I can't prove it in general.

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[*] for example, $\theta_1=(123), \theta_2=(231), \theta_3=(312)$ works, while $\theta_1=(123), \theta_2=(312), \theta_3=(231)$ doesn't.

Answer 1

Your Lemma answers your question.

There is a labelling with $\Gamma_\Theta\le S_n$ if and only if $\Theta$ is equivalent to a left regular representation of itself.

Proof:

$(\Rightarrow)$

$\theta_j(i)=\theta_k(i)\Rightarrow\theta_j=\theta_k$ is equivalent to ${\rm Stab}_\Theta(i)=1$, so $\Theta$ is semiregular. But $|\Theta|=n$ so the orbit-stabiliser theorem implies that $\Theta$ is semi-regular if and only if it is regular.

$(\Leftarrow)$

If $\Theta$ is a left regular representation of itself then $\Gamma_\Theta$ is the right regular representation of $\Theta$ (in particular $\Theta\cong\Gamma_\Theta$).