Non-trivial equivalence relation on $\{1, ..., n\}$ invariant under a simple transitive subgroup of $S_n$

Question

Let $G$ be a simple, transitive subgroup of $S_n$ (meaning that the action of $G$ on $\{1, ...,n\}$ is transitive). Let $\sim$ be a given equivalence relation on the set $\{1,...,n\}$ satisfying the following property: for any $\sigma \in G$,

$$i\sim j \implies \sigma(i)\sim \sigma(j).\tag{$*$}$$

Question: What can we conclude about $\sim$?

Discussion: Obviously we can have the trivial relation. Given a non-trivial $\sim$, I suspect we must have $i\sim j$ for any $i, j\in \{1,...,n\}$.

One can check that $(*)$ implies any $\sigma \in G$ takes an equivalence class $[i]$ to the equivalence class $[\sigma(i)]$. As a consequence all equivalence classes under $\sim$ are of the same size $k>1$. I can construct a homomorphism $G\to S_{n/k}$ by looking at how each $\sigma$ permutes the set of equivalence classes. The kernel of this map is either trivial or all of $G$. In the second case the transitivity of $G$ would imply $n=k$ the desired result.

I am having trouble dealing with the first case. I don't see a problem with $G$ being a subgroup of $S_{n/k}$ where $k>1$. Am I missing something? or is the claim mistaken in the first place?

Note: This question is based on problem 6.5.11 in Berkeley Problem Books in Mathematics.

Edit: Per Derek Holt's answer the claim is indeed incorrect. We know that it is possible to have block size $1<k<n$. What more can we say about this equivalence relation? A question that come to mind is:

• If we are given $k|n$, can we construct a simple transitive subgroup $G\subseteq S_n$ such that there is an equivalence relation $\sim$ with block size $k$?

Answer 1

The equivalence classes form a system of blocks of imprimitivity preserved by $G$. The group is called primitive if the only such equivalent relations are the two obvious ones, where the blocks are either singletons of the whole set.

If you are looking for examples, $A_5$ arises as an imprimitive subgroup of $S_{12}$ with blocks of size $2$, with the action on the cosets of a subgroup of order $5$.

More generally, the action of a group $G$ on the cosets of any proper non-maximal subgroup is imprimitive. More precisely, If $|G:H| = n$ and there is a subgroup $K$ with $H < K < G$ and $|K:H|=k$, then there is an imprimitive action of $G$ of degree $n$ with blocks of size $k$.