The result in the preceding post can be formulated as follows:
A permutation group of degree n = 2k which is transitive on partitions of shape (k,k) but not on ordered partitions of this shape, has a fixed point and is (k−1)-homogeneous on the remaining points, with the single exception of the group PSL(2,5) of degree 6 (with k = 3).
What about groups of degree n = mk, with m > 2 (and k ≥ 2)? Is it true that any group of degree n which is transitive on partitions of shape (k,k,…,k) but not on ordered partitions of this shape has a fixed point?
The character-theoretic argument shows that such a group, if it does not fix a point, must be k-homogeneous. Now of course we could now apply CFSG to say that such groups are known, and simply explore them all. But is there a simpler way?
These questions have some relevance to the theory of transformation semigroups, which I will discuss later.