Let's say that two permutation groups $P_1$ and $P_2$ are isomorphic as abstract groups, but not necessarily permutation isomorphic. How similar will $P_1$ and $P_2$ be, and how much structure will the isomorphism preserve?
Here are some things we know:
- The identity permutation will map to the identity permutation.
- Group theoretic properties, such as order and inverses, will be preserved.
- The cardinality of the set underlying the isomorphism will not necessarily be preserved, even if we require that no element is a fixed point of every permutation.
I guess the flip-side of the question is how different can isomorphic permutation groups be?
EDIT: The reason I ask this question is I'm wondering how feasible it is to study group theory purely in terms of permutation groups. By Cayley's theorem, every group is isomorphic to a permutation group, so it seems like it should be feasible.
This is an example of "how different" isomorphic permutation groups can be. The following subgroups of $S_4$ are all cyclic of order $4$: \begin{alignat}{1} &H_1:=\langle (1243)\rangle \\ &H_2:=\langle (1234)\rangle \\ &H_3:=\langle (1324)\rangle \\ \end{alignat} but the "linearity" condition $\forall i\in\{1,2,3,4\}$: $$\sigma(i)=i\sigma(1)\pmod 5$$ holds for every element of $H_1$, but for no nontrivial one of $H_2$ and $H_3$.