Recently, Michael Giudici, Jesse Lansdown, Gordon Royle and I have constructed a couple of examples of synchronising groups that are radically different from the known examples. You can find the details in our preprint here.

## What is a synchronising group?

The definition came about from the study of synchronising words for finite-state automata. Motivated by the Černý Conjecture on the length of reset words in synchronising automata, Arnold and Steinberg, and independently Araújo, introduced the notion of a *synchronising* group. We say that a permutation group is synchronising if for any non-bijective transformation , the transformation semigroup is synchronising; that is, it contains a constant map (aka. reset word). It was observed in Peter Neumann’s seminal article in 2009 that a synchronising permutation group is primitive. Moreover, the possible O’Nan-Scott types for a synchronising group are heavily restricted. We have just three types: (i) Affine, (ii) Almost Simple, (iii) Diagonal. There are many examples in the first two cases known, but it wasn’t known until now whether the third case was nonempty.

## What was known about synchronising diagonal groups?

Due to the fine work of many people, the shape of a diagonal type group had been whittled down. The *diagonal group* is defined as follows. Firstly, is a nonabelian simple group, and is a positive integer at least 2. The domain for our group action will be the direct product . The group acts *diagonally* on this set in the following action:

We can include inner automorphisms acting identically on each coordinate in , we can have permutations of the coordinates, and also a funny involution (which I won’t describe here). These generate the largest diagonal type primitive group on this domain, and a diagonal primitive group is a subgroup of containing the socle .

Bray, Cai, Cameron, Spiga, Zhang (arXiv, 2018) showed that synchronising diagonal type groups have . Moreover, they showed that “synchronising” and “separating” are equivalent for diagonal type primitive groups. In particular, if as an exact factorisation, then is non-synchronising. For more, see Peter Cameron’s excellent post on this major breakthrough.

## The examples

Take , where . These are the smallest values for which it was not known if the group acting in diagonal action was synchronising or not. (For , and , we have an exact factorisation of . The case was shown to give a non-synchronising example). Then we show that acting on in diagonal action is synchronising. To show this, we needed to show that the action is *separating* and apply Bray, Cai, Cameron, Spiga, Zhang’s result that synchronisation and separation are equivalent for diagonal actions. To show that these groups are separating required us to show that every -invariant undirected graph on has where and are the maximum sizes of a coclique and clique (respectively) of the graph. This in turn needed the theory of association schemes, and in particular, *design-orthogonality*, to reduce the problem to the analysis of just a small number of graphs.