Permutational Isomorphism

Definition

Let $G$ be a group acting on a set $\Omega$ and $H$ be a group acting on a set $\Delta$. Then $G$ on $\Omega$ is permutationally isomorphic to $H$ on $\Delta$ such that for all $g\in G$ and $\omega \in \Omega$, we have $({\omega }^g)\beta =(\omega ){\beta }^{(g)\theta }$. The pair $(\theta ,\beta )$ is called a permutational isomorphism.

Furthermore, if $G=H$ and $\theta =\mathrm{id}$, then we say two actions of $G$ are equivalent.

Theorem

If $G$ and $H$ are both transitive, then they are permutationally isomorphic iff there is an isomorphism $\theta :G\to H$ such that $\theta$ maps a point stabilizer of $G$ onto some point stabilizer of $H$.

In particular, if $G$ and $H$ are both subgroups of $\mathrm{Sym}(\Omega )$ for some set of $\Omega$, then $G$ and $H$ are permutationally isomorphic iff they are conjugate in $\mathrm{Sym}(\Omega )$.

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