A permutation group challenge, 2

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.

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About Peter Cameron

I count all the things that need to be counted.
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One Response to A permutation group challenge, 2

  1. Pingback: Permutation theorem | Jonlineinfo

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