I have done the easier job (linked them from http://www-groups.mcs.st-andrews.ac.uk/~pjc/), but updating and fixing dead links will take longer. Please let me know any problems you find; I will file them away though may not act on them immediately.
Other comments also welcome, of course.
]]>And no, I knew nothing about it until I spent five minutes browsing this afternoon.
]]>Sir Michael Atiyah died yesterday.
I attended part of a course of lectures he gave on algebraic geometry in the 1970s (until term got too busy and I was forced to drop the course). They were excellent lectures, the sort that make you feel you understand everything. (The down side is that, half an hour later, you can’t remember anything, since you haven’t had to work at it.)
The highlight of the lectures was a surprising theorem. Given a “generic” polynomial f of degree 6 over the complex numbers, in how many ways can you write f = g^{2}+h^{3}, where g has degree 3 and h degree 2? Clearly multiplying h by a cube root of unity or g by −1 doesn’t change anything, so ignore this.
The answer is 40. Atiyah proved this by writing down a curve with a horrendous singularity at one point; after dealing with that point, the rest was well-behaved, and he could come up with the answer.
There is a Galois group associated with this situation, of course, which (if I recall correctly) is the finite simple group PSp(4,3). In discussing this, it seemed to me that Atiyah was less sure of his ground. As is well known, he was no great friend to algebra. The Wikipedia article has a quote in which he describes it as the invention of the Devil, which you must sell your soul in order to become proficient in.
I disagreed with him on that, of course.
The picture at the top of this piece is of his portrait in the rooms of the Royal Society of Edinburgh, of which he was president at one time. Indeed, there was a period when he was at the same time President of the Royal Society of London, Master of Trinity College Cambridge, and director of the Isaac Newton Institute. Nigel Hitchen told me that, at this time, if he wanted to have a mathematical conversation with Michael Atiyah, it was necessary for him to stay overnight at the Royal Society and try to catch him at breakfast.
]]>There are a number of results which specifically concern almost simple primitive groups. Notably, there is a classification of these into “large” and “small” groups:
I desribed in the post cited above how we now know that a permutation group which is synchronizing but not separating must be primitive and almost simple. So it is natural to consider first the large groups (since these are well-specified groups with well-specified actions) and look among them for examples of this somewhat elusive class of groups. One infinite family (5-dimensional orthogonal groups over fields of odd prime order, acting on their quadrics) and two pairs of “sporadic” examples, are currently known.
I described here some work on the symmetric groups S_{n} acting on k-element subsets of the domain. In the paper, we give a nice conjecture that, asymptotically (that is, for n large compared to k), this group is non-separating if and only if a Steiner system on n points with block size k exists; by Peter Keevash’s result, this is equivalent to an arithmetic condition on n (it must belong to one of a set of congruence classes).
Leonard Soicher has produced a very efficient program for testing synchronization and separation for primitive groups. For degree 280, a lot of interesting things happen. One special case of this is that S_{9}, acting on partitions into 3 sets of size 3, is non-synchronizing.
Inspired by this, I showed that, if n = kl, with k > 2 and l > 3, then S_{n}, acting on the set of partitions into l sets of size k, is non-synchronizing. The proof goes like this. Take the graph whose vertices are these partitions, two partitions adjacent if they have no common part. This graph is obviously invariant under the symmetric group. I claim that its clique number and chromatic number are equal. To see this, first take the colouring of the graph as follows. Choose an element x of {1,…,n}. For each (k−1)-subset A of the complement, assign colour c_{A} to a partition P if the part of P containing x is {x}∪A. Clearly each colour class is an independent set, so we have a proper colouring. To find a clique with size equal to the number of colours, we use Baranyai’s celebrated theorem (proved using Max-Flow Min-Cut): the k-sets can be partitioned into classes, each of which is a partition of {1,…,n}. Of the resulting partitions, clearly no two share a part, and so they form a clique in the graph.
As explained in the earlier post, having a nontrivial G-invariant graph with clique number equal to chromatic number is equivalent to non-synchronization.
What is of some interest is that, unlike many proofs of non-synchronization, the construction of the colouring is elementary, while the clique requires heavier machinery.
This proof fails for partitions with just two parts, since then the graph constructed is complete. (If two 2-part partitions share a part, they are equal.) Indeed, the group S_{2k} acting on partitions into two parts of size k is 2-transitive (and hence separating) for k = 2 and for k = 3; it is non-synchronizing for k = 4 and for k = 6, by constructions using the Steiner systems S(3,4,8) and S(5,6,12); it is separating for k = 5, shown by computation. So the picture is somewhat unclear!
Continuing, the next class of large almost simple groups is given by classical groups acting on the points of their associated polar spaces. These are non-separating if and only if the polar space contains an ovoid; they are non-synchronizing if and only if the polar space has either a partition into ovoids, or an ovoid and a spread. The question of existence of such structures is not completely settled, despite a lot of work by finite geometers. Note that our one known infinite family of synchronizing but not separating groups are of this type.
What about the small groups? At present we have no good methods for dealing with these. Is it possible that the smallness of base size or order can be used to decide these questions?
]]>A month ago I was engaged in a big fight with a major international academic publisher, whose typesetters had added commas to our paper in such a way as to change the meaning ssignificantly.
Today I found an even more extreme example of this. It was in the Royal Society for the Protection of Birds’ magazine Nature’s Home, in a news article about last summer’s heathland fires in Britain. By using a semicolon rather than a comma in this sentence, they have managed to say exactly the opposite of what they meant.
Please help reduce fire risk on reserves: let fire services know if you spot signs of fire; never light barbecues; drop litter (especially glass) or discard cigarette butts.
]]>Choice woman’s finally made to leave? Hear it’s fallacious (9)
How did it come about that a word with its origin in the Greek for wisdom came to have this meaning?
The sophists were teachers of rhetoric in ancient Athens. Specifically, they taught their students how to argue convincingly in the law courts. As such, they could be honoured as our predecessors in the teaching profession.
In fact, there is an even closer link. Apostolos Doxiadis, in a long article entitled “A streetcar named proof”, in the book Circles Disturbed: The Interplay of Mathematics and Narrative he edited with Barry Mazur, argues that the notion of proof in Euclid is a natural progression from the forensic rhetoric used in the law courts. In the courts, you would argue that if all the alternatives to X (your version of events) are all extremely unlikely, then X is very probably what happened. Euclid uses the stronger version that, if the alternatives to X are logically contradictory, then X is proved. So in some sense the sophists were on the streetcar or tram that carried us to mathematical proof.
So how did they get their bad name?
It seems that Socrates was largely responsible. The sophists’ great sin, in his eyes, is that (like modern teachers) they were paid for teaching; he gave his wisdom away for free in the marketplace (whether his listeners wanted it or not). In fact, he had independent means, and didn’t need payment for teaching. Socrates stands in a pivotal place in the European philosophical tradition; his successors Plato (in whose writings is contained all we know about Socrates’ teaching) and Aristotle set philosophy on its influential path. So Socrates’ views about the Sophists have been accepted largely unchallenged for millennia.
In fact Plato and (especially) Aristotle expressed views on mathematics which were influential but not without problems. Aristotle’s views on infinity (that it was legitimate to talk about potential, but not actual, infinity) and his system of logic kept European thought in a straitjacket which was not escaped without a great struggle.
Indeed, a long report in yesterday’s Guardian suggests that perhaps we have need of a sophist in Britain today. Savage cuts in legal aid resulting from the Tories’ austerity policy have meant that, especially in the family courts, thousands of people cannot afford legal representation and have to represent themselves, leading to numerous miscarriages such as denial of access to their children. Perhaps a television sophist could explain to them how to behave in this situation …
In a further coincidence, here is another of Vlad’s clues yesterday, where he (perhaps unwittingly) made it much easier than it needed to be:
Close to broke in most of country – no alternative to Tory policy? (9)
When I read this, I immediately thought “Tory policy: that means austerity”. On proceeding to justify this from the makeup, I realised that the definition is just “policy”; “Tory” is just there so that when “or” (alternative) is removed from it we get the last two letters of the answer. (The rest is obtained by putting “e” (close to broke) into “austri” (most of country)). But the Guardian report on legal aid makes clear that the cuts have happened since 2010, and so are certainly Tory policy.
]]>
The figure shows two orthogonal Latin squares, the first using the Latin alphabet {A,B,C,D}, the second the Greek alphabet {α,β,γ,δ}, superimposed in the same array. Motivated by this, a pair of orthogonal Latin squares is often called a Graeco-Latin square. Indeed, Euler first defined Graeco-Latin squares (as a technique for constructing magic squares), and then Latin squares, whose name is a back-formation.
A Latin square orthogonal to a given square L is called an orthogonal mate of L.
A source of Latin squares familiar to algebraists consists of Cayley tables of groups. The Cayley table of G is the array whose rows and columns are indexed by the elements of G, with (g,h) entry gh. (Sometimes it is more convenient to take the entry to be (gh)^{−1}; the square obtained is equivalent to the original under a permutation of the symbols of the alphabet.)
So a very natural question is: For which groups G does the Cayley table have an orthogonal mate? This question was considered by Marshall Hall Jr and Lowell J. Paige in 1955. They proved that, if the Cayley table of G has an orthogonal mate, then the Sylow 2-subgroups of G must be either trivial (that is, G has odd order) or non-cyclic. They also conjectured that these conditions are sufficient for existence of an orthogonal mate for the Cayley table of G, and proved this for some groups, including the alternating groups.
A different language is sometimes used here. A complete mapping of G is a bijective function φ from G to itself such that the function ψ given by ψ(g) = gφ(g) is also bijective. If φ is a complete mapping on G, then the Cayley table of G has an orthogonal mate, whose (g,h) entry is gψ(h). (Given x,y in G, we need to find g,h in G such that gh = x and gψ(h) = y; this requires φ(h) = x^{−1}y, from which h and hence g is uniquely determined.) Conversely, the existence of an orthogonal mate implies that of a complete mapping.
In 2009, there was significant progress on the conjecture, in two papers in the Journal of Algebra. Stewart Wilcox, in J. Algebra 321 (2009), 1407–1428, showed that the truth of the conjecture for all groups would follow from its proof for finite simple groups. (Burnside’s Transfer Theorem shows thath a finite simple group, other than the cyclic group of order 2, cannot have cyclic Sylow 2-subgroups, so what is required is to show that all such groups have complete mappings. This is elementary for cyclic groups of odd prime order, so we have to deal with the non-abelian simple groups.) He showed that it holds for all simple groups of Lie type, with one exception, the Tits group (the derived group of ^{2}F_{4}(2)) have complete mappings. Anthony Evans, in J. Algebra 321 (2009), 105–116, handled the Tits group, and also all of the sporadic groups except the Janko group J_{4}. The final sporadic simple group was dealt with by John Bray, but his result remained unpublished.
Now of course it is not entirely satisfactory when the last step in the proof of a long-standing conjecture remains unpublished …
In the meantime, work on synchronization found a use for the truth of the Hall–Paige conjecture. I have described synchronization before, indeed more than once; it is a small obsession of mine now. So here is a very brief summary of all that I need.
So, to test whether G is synchronizing or not, we may assume that G is primitive (this means that it preserves no non-trivial equivalence relation). Primitive groups are described by the O’Nan–Scott Theorem, of which I require here just a simple version: Any primitive permutation group G satisfies one of the following:
The first type are not synchronizing. For such a group preserves a Hamming graph, whose vertices are all the m-tuples over an alphabet of size q, two tuples being adjacent if they agree in all but one coordinate. Such a graph contains cliques of size q (fix all but one coordinate and let the last coordinate take all possible values). It also has a colouring with q colours: take the alphabet to be an abelian group Q, and colour an m-tuple with the product of its entries in Q.
So synchronizing groups must be affine, diagonal or almost simple.
I will not here describe in detail what diagonal groups look like; here are some properties of the “three-factor” diagonal groups. Take a non-abelian finite simple group T. The group in question has minimal normal subgroup consisting of the direct product of three copies of T. To this we can adjoin automorphisms of T (acting simultaneously on the three copies) together with permutations of the three coordinates. This group acts on the set of cells of the Cayley table of G (which we take in the modified form with (g,h) entry (gh)^{−1}, in other words, the unique k such that ghk = 1). The equation ghk = 1 is preserved by cyclic permutation of the three group elements. Now T^{3} acts to preserve these triples. An element t of the first factor acts by right multiplication on g and by left multiplication (by t^{−1}) on h; the other two factors are similar but with the group elements permuted.
This group action preserves the Latin square graph associated with the Cayley table. This graph, defined for any Latin square, has as vertices the cells of the square, with two vertices adjacent if they lie in the same row, or lie in the same column, or contain the same symbol.
Now the clique number of the Latin square graph from a Latin square L of order n is n if n > 2 (a clique consisting of a row, or a column, or a symbol). The chromatic number thus cannot be less than n, and it is not hard to see that it is equal to n if and only if L has an orthogonal mate. (Use the symbols in the orthogonal mate as colours.) Thus, the Hall–Paige conjecture implies that the three-factor diagonal group built from a non-abelian finite simple group is non-synchronizing.
Encouraged by this, I was able to find a proof that diagonal groups with more than two factors are non-synchronizing. The proof for even numbers of factors is elementary, but for odd numbers of factors it uses the Hall–Paige conjecture in a similar manner to the above. The same conclusion was obtained by Pablo Spiga.
With this in hand, I managed to persuade John Bray that the time was right for him to dig out the files of his calculations and publish them. In fact, he did a lot more; he re-did the calculations, and put in several checks on their correctness (enabling different “minimal proofs” to be given). Moreover, he contributed these files to a joint paper proving the result above about synchronization, together with a weaker result covering affine groups and diagonal groups with two factors.
John’s computations are impressive. A result of Wilcox says the following. Let H be a subgroup of a group G. Suppose that H has a complete mapping. Suppose also that there are bijective mappings Φ, Ψ from the set of double cosets D = HxH of H in G such that Ψ(D) ⊆ DΦ(D) for all D. Then G has a complete mapping. To check these conditions, we have to be able to compute the collapsed adjacency matrices for all the orbital graphs for G acting on the cosets of H. (The double cosets correspond naturally to the orbital graphs for G acting on the cosets of H.)
Now the permutation representations of J_{4} have rather large degree. The one he uses, by conjugation on the involutions of type 2A, has degree more than a billion; so only very limited direct computation in this permutation group is possible. The most convenient representation of the group for computation is by matrices of order 112 over the field with two elements. Now since the objects we are permuting are involutions, we can look at their fixed point spaces in this 112-dimensional vector space. John is able to find a “signature” for pairs of involutions, involving four dimensions related to the pair, which distinguish all the 20 orbital graphs.
Computing all the collapsed adjacency matrices would be too big a job. But it is possible to compute those corresponding to the graphs of smallest valency. John computes three such matrices. Now the collapsed adjacency matrices span an algebra over the complex numbers; this is the image of the centraliser algebra of the permutation representation under what is sometimes called the Bose–Mesner isomorphism (in the context of permutation groups it was rediscovered by Donald Higman who notes in his paper that it had also been found by Helmut Wielandt). John verifies that the first two collapsed adjacency matrices generate an algebra of dimension 20, which thus must contain all the others (and they can easily be extracted as the matrices in the algebra having a single non-zero entry in the first row, up to scaling). The third computed matrix then can be used as a check. Now we can check from the collapsed adjacency matrices that all the orbital graphs have triangles, so the condition is satisfied with Φ and Ψ the identity map.
Another check is provided since John finds that all the double cosets apart from H itself have double coset representatives of order 3. These imply the existence of triangles (which are different from the previous in the case when the graphs are directed), verifying the condition required with Φ the identity map and Ψ(D) = D^{−1}.
The paper is on the arXiv, here. As well as the above, it contains weaker results for affine and two-factor diagonal groups, which simplify the computational testing of these groups. For affine and two-factor diagonal groups to be non-synchronizing, it suffices to find a G-invariant graph for which the product of the clique number and independence number is equal to the number of vertices. This result were found independently by Qi Cai and Hua Zhang; so the paper ended up with five authors.
]]>Most of these uses seem completely unconnected. But the use of the same term for the last two is not coincidence, but comes from Galois theory. If L is a Galois extension of a base field E, and K an intermediate field, then L/K is a normal extension if and only if the Galois group of L over K is a normal subgroup of its Galois group over E. (If this happens, the Galois group of K over E is the quotient group.)
But this also hides some mystery. The most important property of normal subgroups is that they are kernels of homomorphisms (and conversely). But step outside group theory, to semigroup theory or universal algebra, and you learn that the kernel of a homomorphism is a partition, not a subalgebra: two elements are in the same part if they have the same image under the homomorphism. It just happens that, in groups, the kernel partition of a homomorphism is precisely the partition into cosets (left or right, it doesn’t matter) of the kernel subgroup.
Indeed, in German, one talks of a “normal divisor” rather than “normal subgroup”, which presumably arises from this interpretation as partition (but I am guessing, I don’t know the etymology).
You can see kernels of homomorphisms in the Galois connection. If L/K is a normal extension, with L Galois over the subfield E, then any E-automorphism of L fixes K setwise, and so induces an E automorphism of K. So we have a homomorphism from Gal(L/E) (the group of E-automorphisms of L) to Gal(K/E). The kernel of this homomorphism consists of the automorphisms which act trivially on K; these are the K-automorphisms of L, the elements of Gal(L/K). [An E-automorphism of L is a field automorphism of L fixing E elementwise.]
In group theory, the term “normal” could be, and sometimes is, replaced by “invariant”. An invariant subgroup is one mapped to itself by all conjugations; this fits in with the notion of fully invariant subgroup, mapped to itself by all endomorphisms. Indeed, for the notion that most people call “subnormal subgroup” (a term in a series of subgroups, each normal in the next, with top element the whole group) was called by Marshall Hall a “subinvariant subgroup”; he remarked in a footnote that he found the term subnormal “unnecessarily distracting”. [Footnote on p.124 of his book The Theory of Groups, published in 1959. He says “The more colorful term subnormal series has been urged on the writer by Irving Kaplansky”, suggesting that it wasn’t yet in common use in 1959.]
All well and good, if a little confusing so far; the first three uses mentioned above are so well separated that probably mathematical papers using each of them form disjoint open neighbourhoods.
But when we come to Cayley graphs, there is real confusion.
Let G be a group, and S an inverse-closed subset of G not containing the identity. The Cayley graph Cay(G,S) is the graph with vertex set G, in which two elements g and h are joined if and only if hg^{−1}∈S. The fact that S is inverse-closed makes the graph undirected, and the fact that it doesn’t contain the identity makes the graph loopless. The group G acts on itself by right multiplication; this action embeds G into the automorphism group of the Cayley graph.
Now each of the following two definitions occurs in the literature:
These two definitions are quite different. Indeed, the second one restricts the symmetry of the graph (its automorphisms are all contained in the normaliser of G in the symmetric group), while the second expands it (the left, as well as the right, action of G consists of automorphisms).
The complete graph on G is a Cayley graph for any group G; it is normal in the first sense but not the second (if the order of G is greater than 4). On the other hand, the Cayley graph of S_{3} with respect to two of its transpositions is a 6-cycle, and its automorphism group contains S_{3} as a (normal) subgroup of index 2; so it is normal in the second sense but not the first (since the three transpositions are conjugate).
Both terms, as I said, are well-established, and it is probably too late to change the terminology now.
This was on my mind because of recent events. The argument about synchronization for groups with regular subgroups mentioned in the last-but-one post depends on a relevant graph being a normal Cayley graph (in the first sense); but I learned about the result of Cai and Zhang at the conference in Shenzhen, which also had a talk about normal Cayley graphs (in the second sense).
Philosophers of mathematics argue about whether mathematics is discovered or invented. In the book of Genesis we read that God created the animals but Adam gave them their names. I think what the examples above show is that, whether mathematics is discovered or invented, the names we give to the concepts are our own invention.
]]>Sometimes the weather at this time of year can be beautiful, and the Reading campus looking lovely with autumn leaves under blue skies and a variety of waterfowl on the lake. But this time it was wet; Rosemary and I started out round the lake when the rain seemed to have stopped, but were thoroughly drenched by the time we got back.
The highlight of the meeting was a talk by Imre Leader on new work he and Paul Russell have done on partition regularity for inhomogeneous equations, going right back to Rado’s work at the beginning of the subject. A system Ax = b over the integers is partition regular if, in any finite colouring of the natural numbers, you can find a monochromatic solution to the equations. We are interested here in the inhomogeneous case, where the vector b is non-zero. In this case, if the system has a constant solution (with the values of all variables equal), it is clearly partition regular. If s denotes the column vector which is the sum of all the columns of A, then this holds if and only if b is a multiple of s (say b = as, in which case putting all variables equal to a gives a solution).
Rado proved the converse, that is, if the system is partition-regular then there is a constant solution. Imre began by taking us through the proof. First he did the case where there is just one equation. Then rather complicated arguments do the general case. Various authors since have extended this, using algebraic techniques of increased sophistication, to extend from the integers to other rings (so the result holds for integral domains, and for Noetherian reduced rings).
Remarkably, what Imre and Paul have done is to handle all commutative rings with identity, not by elaborating the existing arguments, but by going back to Rado’s original proof for one equation and extending it to any number. The trick is that, although we work over a ring R, we are going to think of it in the proof as an abelian group and essentially ignore the multiplication after the very first step. So here is the proof.
We suppose that there is no constant solution. Let K be the subgroup of R^{n} consisting of all multiples of s. By assumption, b is not in K. Now let H be a subgroup of R^{n} which is maximal with respect to containing K but not b. Then R^{n}/H is an abelian group containing a non-zero element (the image of b) which lies in every non-zero subgroup (if not, we could replace H by a larger subgroup). Now such abelian groups are not too hard to characterise: they consist of finite cyclic p-groups and Prüfer p-groups, for some prime p. The latter consists of all the p-power roots of unity in the complex numbers; so, in either case, the group can be represented on the unit circle. Now dividing the circle into sufficiently small intervals allows the construction of a colouring with no monochromatic solution of the original system.
After this lecture, Anthony Hilton fetched in from the common room the bust of Richard Rado from the common room next door, where it sits unnoticed in a corner (despite Anthony’s attempts to have it more prominently displayed).
After that, a briefer account of the rest.
I was the first speaker. I talked about equitable partitions of Latin square graphs, a topic I have described here. Partly inspired by some work I am currently doing with Sanming Zhou from Melbourne, I introduced the topic with a few words about perfect 1-codes in graphs. I sometimes think that, had we mentioned this in the introduction of the paper, it might not have been rejected without refereeing by a journal with the words “designs” and “codes” in the title.
Dudley Stark talked about a theorem he and Nick Wormald have spent the past twenty years proving: estimates for the probability that a random graph G contains a fixed graph H. They handle both the G_{n,m} and G_{n,p} models (the answers are slightly different), and they are able to extend the range of parameters previously handled (even in the case where H is a triangle). The idea is quite simple, but the details extremely complicated. Dudley said they never doubted that the result was true but were not sure whether they would be able to produce a proof.
Richard Mycroft talked about the function t(T), where T is an edge-oriented tree on n vertices, defined to be the smallest N such that T is embeddable in every N-vertex tournament. It was conjectured by Sumner that 2n−2 vertices suffice; this is known for large n (“regularity lemma large”) and small n (up to 8, maybe), and thought to be true for intermediate values. A tree T is called unavoidable if t(T) = n; Richard and his colleague have shown that almost all trees are unavoidable.
Peter Larcombe talked about some rather different mathematics, involving Catalan numbers and Catalan polynomials. This touched on Newton–Raphson and other approximation methods including Padé approximants, and he also mentioned some new results about powers of 2×2 matrices (and more generally, tridiagonal matrices) which he has observed and proved, challenging us to say if we had seen such things before (no one had). It all went by rather fast; I hope to say more on some later occasion.
The final talk was by Matthew Johnson, who talked about almost bipartite graphs, those whose vertex set can be partitioned into an independent set and a set inducing a forest. It is known that a graph of maximum degree 3 is almost bipartite; Matthew and his colleagues have found a different proof of this which gives a linear-time algorithm for finding the partition. The application is to a recolouring problem. Given two proper colourings of a graph, with the same set of colours, is it possible to move from one to the other by changing the colour of one vertex at a time (retaining the properness)? In some cases they have a simple algorithmic answer to this question.
]]>Here is a brief recapitulation.
Let G be a transitive permutation group on Ω, where |Ω| = n.
The “official” definition of synchronization hearks back to the origin of this concept in automata theory; but what is said above will do for now. Note that if G is non-synchronizing, then A and any part of P witness the fact that G is non-separating; hence separating implies synchronizing.
For the record, I also mention the graph-theoretic interpretation of these concepts:
Now a synchronizing group is primitive (i.e. preserves no non-trivial equivalence relation), as we can see from the graph-theoretic interpretation by observing that an imprimitive group preserves a complete multipartite graph. Also, a synchronizing group is basic (in the O’Nan–Scott classification), since a non-basic graph preserves a Hamming graph. So O’Nan–Scott tells us that such a group is affine, diagonal or almost simple.
While I was in Shenzhen earlier this month, I was given a preprint by Qi Cai and Hua Zhang showing that, for groups of affine type, synchronization and separation are equivalent. Just before I left, Mohammed Aljohani had told me that the same is true for groups of diagonal type with two socle factors.
There is in fact a common approach to these two results. Suppose that the group G has a regular subgroup H. This means that there is a unique element of H mapping any given point of Ω to any other. Choosing a fixed element α of Ω to correspond to the identity, we can let the element αh correspond to h, for any h in H. Thus we identify Ω with H. Now the following two propositions are straightforward to prove:
Proposition 1 If A and B witness non-separation, then A^{−1} and B give an exact factorisation of H (this means that any element of H is uniquely expressible as xy with x in A^{−1} and y in B).
Proposition 2 If A and B give an exact factorisation of H, then the sets Ab for b in B form a partition of H, and this partition and the subset B witness non-synchronization.
These results come close to showing that synchronization and separation are equivalent for groups with regular subgroups; but there is that pesky business about the inverses which I can’t get around in general. But there is a special case where everything works fine. Note that a G-invariant graph must be a Cayley graph for the subgroup H (admitting the right action of H as automorphisms). If any such graph also admits the left action of H, then A being a clique implies that A^{−1} is a clique, and so the gap is closed. This is the case if G contains both the left and right actions of H, which is true in the 2-factor diagonal case. It is also true in the abelian case, since these actions are then the same. But in a group of affine type, the translation group is an abelian regular subgroup; so we recover the above special cases.
It is also the case that diagonal groups with k factors in the socle, for k > 2, are necessarily non-synchronizing. This, it turned out, had already been proved by Pablo Spiga, but the manuscript had been lost, so I had to re-do his arguments myself. I do not want to go through it here, since defining the diagonal groups is a moderately complicated business; but I want to mention a remarkable feature of the proof.
There is a natural graph associated with a diagonal group, which is the union of k overlapping copies of a Hamming graph, or can be regarded as a Hamming graph with some “diagonal edges” added. This is closely related to some work Cheryl Praeger and Csaba Schneider discussed at the Shenzhen conference. It is possible to show that, for a diagonal group with k simple factors T in the socle, where k > 2, this graph has clique number and chromatic number equal to |T|.
For the clique number, this is easy to see; the cliques are visible in the Hamming graphs (which have dimension k−1 over the alphabet T). The trick is to show that the chromatic number is also equal to |T|, by colouring the graph using T as the set of colours. For k even, there is a simple direct rule for the colouring; but for k odd, this requires the truth of the Hall–Paige conjecture, fairly recently proved.
A complete mapping on a group G is a bijection φ from G to itself with the property that the map x→xφ(x) is also a bijection. The latter map, which I will call ψ, is also called an orthomorphism of G.
The existence of a complete mapping has the effect of producing a Latin square orthogonal to the Cayley table of G. And now we see the relevance to non-synchronization. For the case of three factors, the graph referred to above is just the Latin square graph associated with the Cayley table of G (the vertices are the cells, and are joined if they are in the same row or column or have the same entry), and a Latin square orthogonal to the Cayley table gives a proper colouring of this graph with |G| colours.
Hall and Paige conjectured that a group has a complete mapping if and only if its Sylow 2-subgroups are not cyclic. They proved the necessity of the condition, and its sufficiency for alternating groups. Wilcox reduced the proof of the conjecture to the case of simple groups, and dealt with groups of Lie type except for the Tits group. Evans did this group and also all the sporadic simple groups except for J_{4}, which was handled by Bray. Thus the conjecture is proved, the only small problem being that John Bray has not yet published his proof.
Groups which are synchronizing but not separating are very rare. We have one infinite family of examples, and one sporadic example, and that is all.
But these recent developments show that such a group must be almost simple. So at least we know where they are hiding!
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