Sean Eberhard commented on my posts on diagonal groups (see here and here). He is correct; there is an association scheme preserved by the full diagonal group with *n* factors in the socle; it is non-trivial if *n* > 2. The details take a few pages to write out but the basic idea is completely fine.

This shows that an *AS-free group* (one which does not preserve a non-trivial association scheme) must be either 2-homogeneous or almost simple. Clearly every 2-homogeneous group is AS-free; there are almost simple examples, but they are rather strange and no pattern has emerged thus far.

I hope a preprint will go on the arXiv sometime soon. But you read it first here (in Sean’s comments over the last few days).

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About Peter Cameron

I count all the things that need to be counted.

The paper is now on the arXiv: https://arxiv.org/abs/1905.06569 .

As a bonus, we construct an association scheme (a refinement of the above) from the Latin hypercube consisting of all

n-tuples of elements of a groupGwith product equal to the identity. This was previously known only for abelian groups ifn>2.