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

Let be a simple group and let be the corresponding diagonal group acting on . Suppose acts transitively on some set . Let be the set of all derangements in this action, and consider the Cayley graph on generated by . Assume that is a characteristic subset. There is an obvious -colouring: colour according to the destination of any fixed .

The big question is whether there is an -clique. This is certainly the case whenever has a regular subgroup: this settles the question for lots of small examples. Sometimes there is an -clique for other reasons, too. For with its defining action, the existence of of an -clique is equivalent to the existence of a Latin square all of whose rows are even (I suppose these always exist, but I don’t immediately see a construction when is twice an odd number).

]]>There is a nice graph invariant under the diagonal group with any number of factors, which I used to show that these groups are non-synchronizing. I thought at first that it would be a “Latin hypercube graph”, analogous to the Latin square graph for 3 factors. (The Latin hypercube being constructed as the set of all k-tuples with product 1 in the group.) But these graphs are not the same, and the one built from the hypercube does not admit the full diagonal group except in the case of 3 factors.

There is some subtlety here I think. ]]>

Actually, maybe that doesn’t work because the graph is not strongly regular: it’s not enough to know just whether there are such indices ; we need to know how many, and how they’re arranged. So for each partition of the indices , define relation by if and only if the partition defined by coincides with . Does this work?

In the case we’re saved by the conjugacy classes thing, at least for finite groups. (Incidentally I think we need to combine each conjugacy class with its inverse to get symmetry.) On the other hand if is an infinite group with just two conjugacy classes then the corresponding diagonal group is 2-transitive!

]]>On the face of it, there is nothing absolute about mathematical theories; they are created by mathematicians in a similar way to the way that God is created by religious people. But the astonishing things, which any philosophy of maths has to address, are: firstly, mathematics is consistent and stable over long periods of time; and secondly, our inventions are “unreasonably effective” in describing the real world. For example, Newton and Leibniz invented calculus, which relies on infinitesimally small quantities, which clearly don’t exist in the real world; and yet calculus underpins a huge amount of modern life via its applications in science and technology. It is not inconceivable to me that the same might be true in the religious context; let me just say, I have yet to be convinced of this. ]]>

The power on my laptop ran out. Luckily we made it to the end of the talk, but only just. We missed any questions/conversation afterwards unfortunately.

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