This question is about Problem 6.5.12 in the Berkeley Problems in Mathematics, 3rd edition, which reads as follows:
Let $G$ be a transitive subgroup of the group $S_n$. Suppose that $G$ is a simple group and that $\sim$ is an equivalence relation on $\{1,...,n\}$ such that $i\sim j$ implies that $\sigma(i)\sim \sigma(j)$ for all $\sigma \in G$. What can one conclude about relation $\sim$?
After working through the problem I'm still not quite sure what answer this problem might be looking for (unfortunately the answer is not in the back of my edition). According to Keith Conrad's posting "Transitive Group Actions", relations such as the above are called $G$-equivalence relations, and they can be classified as follows: fix an index in $\{1,...,n\}$ (let's just say $1$ for definiteness). Then let $H$ be the stabilizer of $1$. The $G$-equivalence relations on $\{1,...,n\}$ are in bijective correspondence with intermediate subgroups $H \subset K \subset G$ (Theorem 7.5 in the aforementioned article).
All this sounds nice and reasonable to me - but I don't see how simple groups are relevant in any of this, except to say that if $G$ is abelian and simple then there are no non-trivial such equivalence relations since there are no non-trivial intermediate subgroups. Is there an answer to the above question that takes into account the "simple" part of the requirement in a natural way?