The result in the preceding post can be formulated as follows:

A permutation group of degree *n* = 2*k* 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.

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About Peter Cameron

I count all the things that need to be counted.

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