The numbers must all be multiples of 3, since the colouring has all colour classes of equal size. But all numbers above are odd, and all except 21 are multiples of 9; I don’t know why this is.
How does the sequence go on? It contains 243, 441, 729, … but I don’t know if there are other terms before these.
We say that a permutation group G on the set {1,…n} synchronizes a non-permutation f from this set to itself if the semigroup generated by G and f contains a constant map. Also, the kernel classes of a map f are the inverse images of the points in the image of f; and f is uniform if all the kernel classes have the same size, and non-uniform otherwise.
A group that synchronizes every non-permutation must be primitive. It is not true that a primitive group synchronizes every non-permutation; there are some very interesting exceptions, and deciding the question is very difficult. The current “big conjecture” on synchronization for primitive permutation groups is due to João, and says:
Conjecture: A primitive group synchronizes every non-uniform map.
I have spoken about this conjecture in several places. We had proved it for maps of rank at most 4 (where the rank is the cardinality of the image) or at least n−2, in a paper published this year in the Journal of Combinatorial Theory Series B (doi: 10.1016/j.jctb.2014.01.006). We are currently hard at work improving the upper bound, and hope to get it down to n−4 or even n−5.
This morning, we discovered that the conjecture is false.
Here is a brief description of the counterexample.
The graph in question is the line graph of the Tutte–Coxeter graph, also known as Tutte’s 8-cage. The Tutte–Coxeter graph is trivalent; its line graph has valency 4, and any edge lies in a unique triangle (the triangles corresponding to the vertices in the original graph), so the closed neighbourhood of a vertex is a butterfly, consisting of two triangles with a common vertex.
As in dynamics, it is the butterfly that causes chaos with a flap of its wings …
Our graph has automorphism group G = PΓL(2,9), and has chromatic number 3; this means that it has a homomorphism onto one of its triangles, this being a (uniform) map of rank 3 not synchronized by the primitive group G.
We used GAP and its share package GRAPE to determine all the independent sets of size 15 in the graph (up to the action of the group G, there are just two of these), and to examine the induced subgraph on the complement of each such set. In both cases, this induced subgraph turns out to be a disjoint union of cycles of even length; so each independent set of size 15 is a colour class in a 3-colouring. For one of these independent sets, it occurs that the induced subgraph on the complement has two components, of sizes 10 and 20. In this case, the original independent set A and the bipartite blocks B and C of the component of size 10 and D and E of the other component are all independent sets, with edges between A and the others, and between B and C and between D and E, and no further edges.
Thus the edges between these five sets can be mapped homomorphically to the butterfly, with A mapping to the central vertex. This gives us a non-uniform map of rank 5 (with kernel classes of sizes 15, 5, 5, 10, 10) which is an endomorphism of the graph, and hence not synchronized by the group G.
Indeed, this butterfly resembles Lorenz’s attractor in one respect. If you wander round the graph and follow your image under the homomorphism, it will move around one wing of the butterfly and then (apparently randomly) switch to the other wing, and so on. Actually, since the wings are of unequal sizes, you will spend most of your time on the larger wing.
Is this an isolated example, or the first butterfly of the summer? (It is not completely isolated; the line graph of the Biggs–Smith graph gives another example with degree 153, where the kernel type is (6,6,45,45,51).)
As with any good counterexample, it opens various new questions. Is it the smallest counterexample? What can we say about the gap between the ranks of the smallest map and the smallest non-uniform map not synchronized by a primitive group? (It can’t be 1; how large can it be?) Does a primitive group of degree n synchronize every map of rank greater than n/2? Can one determine all the primitive groups which fail to synchronize some map of rank 3? And so on …
This group was set up by Carrie Rutherford, whose photo is below (as well as a mathematician at South Bank, she is a volunteer on the Markfield beam engine near Tottenham).
Carrie learned how to run a study group as a member of the Combinatorics Study Group at Queen Mary. She set up her own at South Bank, and it has evolved its individual style, attracting a range of people from as far afield as Norwich and Portsmouth.
The first ever talk in the MSG was on Hadamard matrices, and when Carrie asked me if I could talk about Hadamard matrices I was happy to comply.
Hadamard asked the question:
How large can the determinant of an n×n real matrix be, if its entries all have absolute values at most 1?
There is a simple geometric argument. The determinant is the volume of the Euclidean paralleleliped spanned by the rows of the matrix. The Euclidean length of each row is at most n^{1/2}, and the volume spanned by vectors of fixed length is maximised when the vectors are pairwise orthogonal; so the maximum determinant is n^{n/2}. Equality is achieved if and only if all the entries in the matrix H are +1 or −1 and HH^{T} = nI. Such a matrix is called a Hadamard matrix. (This is a nice example of how the solution to a continuous problem may plunge you into the discrete world.)
It is easy to show that the order of a Hadamard matrix must be 1, 2 or a multiple of 4. It is conjectured that every multiple of 4 is the order of a Hadamard matrix. This is one of the big open problems in discrete mathematics. I think that the first unsettled case is order 668. (The previous smallest, 428, was solved at the IPM in Tehran shortly after my visit there in 2005.)
There has been some interest in symmetric and skew Hadamard matrices. (A Hadamard matrix can’t really be skew, since a real skew matrix has zero diagonal; we abuse language by saying that a skew-Hadamard matrix is one with +1 on the diagonal and H−I skew-symmetric.)
One could ask Hadamard’s question also for matrices with zero diagonal and off-diagonal entries with absolute value at most 1. The same argument shows that the determinant of such a matrix C is at most (n−1)^{n/2}, with equality if and only if the off-diagonal entries of C are +1 or −1 and CC^{T} = (n−1)I. A matrix meeting the bound is called a conference matrix. (The name comes from their occurrence in conference telephony in the 1950s.)
One of the most remarkable facts about conference matrices is that, essentially, such a matrix must be either symmetric or (genuinely) skew-symmetric:
Theorem: A conference matrix of order greater than 1 has even order, and is equivalent (under row and column sign changes) to a symmetric matrix if n is congruent to 2 (mod 4), or to a skew-symmetric matrix if n is congruent to 0 (mod 4).
It is easy to see that C is a skew conference matrix if and only if C+I is a skew-Hadamard matrix, so the existence conjecture in this case is that both types exist for all multiples of 4. In the symmetric case, however, van Lint and Seidel showed that a symmetric conference matrix of order n exists if and only if n−1 is the sum of two squares.
Symmetric conference matrices are obtained by bordering with 1s the Seidel adjacency matrices of Paley graphs. Now the South Bank University logo is a pentagon, and the Mathematics Study Group logo the 3×3 grid; these are the first two Paley graphs, so give rise to conference matrices of orders 6 and 10, the latter very fitting for the anniversary of the MSG!
A similar construction shows that skew-Hadamard matrices are obtained by suitably bordering the signed adjacency matrices of doubly regular tournaments. By coincidence, these arose in another piece of work I put on the arXiv recently, joint with statisticians from the UK, Germany and Poland on circular repeated-measurements designs.
Also from statistics is an intriguing conjecture by Denis Lin on the relation between maximum determinant of skew matrices with orders congruent to 2 (mod 4), similar to the relation H = C+I that we saw in the case where the order is congruent to 0 mod 4. However, for this, I will refer you to the slides, which are on the on the MSG webpage, or in the usual place.
Maybe it is different this time …
Last Tuesday I addressed the newly revitalised Queen Mary Mathsoc, at the invitation of Giulia Campolo. Perhaps she brings Italian flair to the job; in any case, she had produced a T-shirt with a logo devised by an Italian designer, and the result is impressive (if you can ignore the model).
I particularly like the choice of logo, which is described on the back as
follows:
A humble reminder of the frustration of all mathematicians, their effort and dedication in pursuit of a new solution, a breakthrough formula, or a lifetime chimera.
A dimension for human speculation and a place for abstraction, where both victory and failure are a possibility.
The second and third year students took my Mathematical Structures course. I don’t claim that I taught them the appreciation of mathematics which this logo and its description show; but I hope I contributed in some degree.
Anyway, I really enjoyed the evening. I had an audience of close to 100; this included first-year students as well as students I had taught, as well as others from computer science, physics, even genetics. I talked from 6 till 7, and afterwards we sat around and chatted, and it wasn’t until 9 that I noticed it was getting late and I should go.
I talked about Paradox. I wanted to get across my view, which is that rather than the famous paradoxes (infinitesimals, Russell’s paradox, Gödel’s Theorem, and the rest) being destructive of mathematics, they greatly enrich it by showing new aspects of our playpen, much as non-Euclidean geometry did. The three examples above gave us calculus, axiomatic set theory, and non-standard arithmetic and analysis.
Added 13 October: Here is a picture taken after the lecture by Martyna Sikora:
In late May, I was in Hay-on-Wye at the How the Light Gets In festival.
I talked about humanity’s engagement with infinity over the last few millennia, from Malunkyaputta’s questions to the Buddha and Aristotle’s disavowal of a completed infinity to Cantor, Hilbert and Gödel in the twentieth century. I was very pleased with the way it went, and later in the day met some of the audience discussing it.
The course was filmed, and has just gone live on the website of the IAI Academy. (The initials are for the Institute of Arts and Ideas, which runs the festival and now provides on-line courses from its Academy.) You can find it at http://iai.tv/iai-academy/courses/take/home?course=the-infinite-quest.
You have to register to take the course, but registration is free. There is additional material available, and the possibility of adding more; there is a discussion forum; and you earn a certificate if you complete the course.
I think they have done a great job, given that the lecture was filmed in a tent in the middle of a very muddy field. Take a look if you are interested. When I talked about infinity on “Horizon” on the BBC it provoked a lot of discussion and comment, and I hope that this does too. Indeed, one of the hardest parts of preparing the course material was coming up with questions with a “right answer”: the first few I suggested were more like invitations to discuss things.
In the last two posts on regular polytopes, I gave away something about my method of working. Although I have known about regular polytopes for a long time, I have never attempted to do research on them before. I find the best way to start on a new project like this is to explain it to myself, and I took the opportunity of posting my explanations in case they were helpful to others.
Anyway, the attempt worked: during my time in Auckland (now, alas, over), Dimitri Leemans and I did make some progress on this. I would like to explain now what we did, in context: I will begin by describing what Dimitri and his co-authors Maria Elisa Fernandes and Mark Mixer did, and then go on to our new results and how they fit in.
By the way, the weather wasn’t always as pleasant as in the photo above!
To begin with a reminder: a regular polytope whose automorphism group is a group G is equivalent to an expression for G as a string C-group of rank d: this means that G is generated by d involutions (elements of order 2), say ρ_{0}, … ρ_{d−1} with the properties
This can be rephrased as follows: the map I→G_{I} from subsets of {0,…d−1} which takes each subset to the subgroup generated by the corresponding involutions is an embedding of the Boolean lattice of rank d into the subgroup lattice of G, with the properties that the atoms map to subgroups of order 2 and the rank 2 elements corresponding to non-consecutive indices map to subgroups of order 4.
Note that we do not insist that adjacent involutions fail to commute. (If they do commute, then the polytope is in a certain sense “degenerate”, for example, a face might have just two vertices and two edges.)
The diagram associated with the string C-group has vertices labelled 0,…d−1, two vertices joined if the corresponding involutions do not commute. It is thus a subgraph of the “string” with consecutive pairs joined. Missing edges correspond to neighbouring involutions which commute.
An advantage is that, if the diagram is not connected, then the group is the direct product of the subgroups corresponding to the connected components; so for classification problems we can restrict to the connected case.
The main benefit of this convention is that it is ideal for induction: any subset of the generators of a string C-group themselves generate a string C-group.
The basic question we want to address is:
Problem: Given a group G, what can be said about the regular polytopes with automorphism group G, and in particular, what is the maximal rank of such a polytope, and what polytopes have rank equal to or close to this bound?
It follows from what was said earlier that, if the group G is not decomposable as a direct product, then the diagram must be connected.
A very natural group to start with is the symmetric group S_{n}. This is the automorphism group of the regular (n−1)-simplex: the equilateral triangle for n = 3, the regular tetrahedron for n = 4, and so on. In this case, the involutions are the Coxeter generators for the symmetric group: ρ_{i} is the transposition (i+1,i+2) for i = 0,…n−2.
The generators of a string C-group form an independent set in the group: none lies in the subgroup generated by the others. As I explained, Julius Whiston showed that an independent set in S_{n} has cardinality at most n−1, and Philippe Cara and I showed that a string C-group is obtained only in the case when the elements are the Coxeter generators. So the regular (n−1)-simplex is the only regular polytope of rank n−1 with automorphism group S_{n}.
Leemans and his co-authors found a remarkable extension of this result, which is not yet published; Dimitri gave me a preprint of the paper to learn about what they had been doing at the start of my time in Auckland.
Fernandes and Leemans proved that, for n ≥ 7, there is a unique polytope of rank n−2 with automorphism group S_{n}. The three authors conjecture a wide generalisation of this:
Conjecture: There is a function f on the positive integers with the property that, if n ≥ 2k+3, then there exactly f(k) distinct polytopes of rank n−k with automorphism group S_{n}.
They proved the conjecture for k = 3,4, with f(3) = 7 and f(4) = 9. Computation suggests that f(5) = 35. I am sure you find these numbers as intriguing as I do!
Fernandes, Leemans and Mixer constructed regular polytopes of rank ⌊(n−1)/2⌋ with automorphism group A_{n}, and conjecture that this is best possible when n > 11. (For n = 11, there are rank 6 polytopes.)
From the earlier comments about induction, it is clear that if we have a regular polytope whose automorphism group is S_{n} or A_{n} (equivalently an expression for either of these groups as a string C-group), then any subset of the generators will give a string C-group which is a subgroup of S_{n}.
So, to attack either of the questions above, we must look more generally at expressions for arbitrary subgroups of S_{n} as string C-groups.
We know that primitive groups not containing the alternating group have small order. This can be proved without the Classification of Finite Simple Groups, but the strongest results are obtained using CFSG. The best result is that of Attila Maróti. Such a group satisfies one of the following:
We can make strong statements in each case.
In the first case, we can replace the given action of G on n points by an action on a much smaller set (either G is S_{m} or A_{m} acting on m points, or in the wreath product case we can take the imprimitive action of G on ml points). In the first case we use induction; in the second, use the analysis of imprimitive groups below to bound the rank.
The Mathieu groups are handled by direct computation. (Dimitri has a Magma program which will find all polytopes with a given, not too huge, group as automorphism group.)
In the third case, we confront the upper bound above with any one of several lower bounds:
These arguments give bounds much smaller than n/2 for the rank, except in a few small cases.
Here we were able to find the best possible bound.
If n is even, then the cross-polytopes (the series beginning with the square and the regular octahedron) give examples with rank n/2. I started trying to prove that this case had the largest rank; I failed because there is sometimes a polytope with rank one greater. Our final result reads as follows.
Theorem: A regular polytope whose automorphism group is a transitive but imprimitive subgroup of S_{n} has rank at most 1+n/2; equality is realised only if n is congruent to 2 (mod 4), in which case there is a unique polytope with this rank.
These arguments show that, in order to prove the conjectured upper bound for the rank of regular polytopes with group A_{n}, we can suppose that the “maximal parabolic” subgroups (generated by all but one of the involutions) are subgroups of S_{n} consisting of even permutations, so our analysis above applies. Moreover, such a subgroup, if transitive, is imprimitive (except in small cases).
We spent some time looking at the two-orbit case. Suppose the orbit lengths are n_{1} and n_{2}. An ingenious argument due to Dimitri shows that, if the group on the first orbit is the symmetric group, then its rank is at most about half its degree. A similar statement holds if it is alternating (by induction), primitive (and usually we can do much better), or transitive imprimitive (by our theorem). So we are quite close to being able to prove the conjecture!
That is enough for now, I think.
Last weekend, as I mentioned, we took a long weekend and did some travelling, at the kind invitation of Hans Hockey.
On Saturday morning we took the bus to Hamilton. Hans met us at the bus station, drove us round the lake, and took us to the jetty for a lunch cruise on the Waikato river. After that, we had a leisurely stroll round the Hamilton Gardens, one of my favourite places: it features beautifully executed gardens from various places and periods including Japanese meditation garden, Chinese scholar’s garden, West Coast modernist garden, and Italian renaissance garden.
It was the 50th anniversary of the University of Waikato, and we went to the Staff Revue, a pantomime of Little Red PhD Hood and the Wicked Librarian, interspersed with musical interludes, mostly music from 1964 (my first year as a university student and a great year for music!)
The next morning we drove with Hans’ family to their house in Taupo. In the afternoon we climbed the 1088 metre Mt Tauhara, overlooking the lake. We soothed tired muscles with a dip in the hot mineral waters.
On Sunday, we drove to Napier. The spectacular mountain views were somewhat wasted on us, as for most of the drive we had rain and fog.
Napier was flattened by an earthquake and fires in 1931, but was quickly rebuilt, mostly in the Art Deco style popular at the time. Much of the Art Deco survives and is one of the town’s claims to fame now. Unfortunately, our views of this were also hampered by the pouring rain. After sightseeing, while Hans and family visited his sister, we drove back to Taupo.
The next morning, the clouds over Taupo finally cleared and gave us the view at the top of this post, of Ruapehu seen across Lake Taupo. This vast lake takes up part of the caldera of a huge volcano that erupted in about 200AD, possibly affecting climate and crops over much of the northern hemisphere.
We drove back to Hamilton and went to the University of Waikato, where our host was Nick Cavenagh. We had lunch with the mathematicians and statisticians, and then gave back-to-back talks, before catching the bus back to Auckland.
When we arrived, Hans showed me a picture he had taken, of the 81 SET cards laid out on a table in an arrangement with a number of remarkable properties, which I describe here. It turns out that it is a picture of an isomorphism between the SET pack and the Sudoku grid.
The game SET is played with a pack of 81 cards. Each card carries a picture
with four attributes, each with three possible values:
A set of three cards is a SET if the values of each attribute are either all the same or all different. It is easy to see that any two cards are contained in a unique SET; so the cards and SETs form a Steiner triple system of order 81.
Label the three values of each attribute with the three elements 0, 1, 2 of the field GF(3) (the integers modulo 3). The labelling is arbitrary, although it is natural to label the number of symbols with its value modulo 3. Thus each card is identified by a 4-tuple of elements of GF(3).
The 4-dimensional affine space over GF(3) is defined as follows: the points are the vectors of a 4-dimensional vector space over GF(3); lines, planes and 3-spaces are cosets (translates) of subspaces of dimension 1, 2, 3 respectively of the vector space.
For example, a line has the form {0,b,2b}+a = {a,a+b,a+2b}, where a and b are vectors and b ≠ 0. Observe that
As a corollary of the last two statements, we see that three cards form a SET if and only if their coordinates form a line in the affine space. So we have identified the Steiner system formed by the SET cards with the points and lines of the affine space.
Three points of the affine space which do not form a line “generate” a plane. (If the three points are x,y,z, then the plane is the translate by x of the vector subspace spanned by y−x and z−x.)
The structure of the 9-point affine plane over GF(3) is most easily shown by arranging the points in a 3×3 array, so that the rows and columns of the array are lines. The other lines can be described as the sets of three positions corresponding to the six terms in the expansion of a 3×3 determinant: that is, each set has one point in each row and one in each column, and the six possible such sets form the remaining lines. In particular, the two diagonals of the square are lines.
This can be realised by SET cards as shown.
In a similar way, four points, with no three collinear and not contained in a plane, generate a 3-space, which could be imagined by going into the third spatial dimension and stacking three planes one above the other. Then the vertical and diagonal lines containing three points, one from each plane, are also affine lines; but there are others too, which become harder to imagime.
Five points, with no three collinear, no four coplanar, and not lying in a 3-space, generate the entire geometry.
We’ve seen how to lay out nine cards forming a plane.
Now choose one of the 72 remaining cards and put it next to the plane, in row 1 and column 4. Then we can fill in three rows of nine cards each to form a 3-space. For example, positions (1,1), (1,4) and (1,7) must form a SET, which determines the card in position (1,7).
Finally, choose one of the 54 remaining cards, and place it in row 4 and column 1. Now similar rules determine the position of every card in the pack. We end up with a layout which might look like this:
The layout has various other properties. For example, the central card in the centre square and any two cards in diametrically opposite positions always form a SET.
The 9×9 layout can be matched up with the Sudoku grid. Details are in a paper by Rosemary Bailey, Robert Connelly and me in the American Mathematical Monthly 115 (2008), 383–404.
We coordinatise the grid with four coordinates, each taking values 0, 1 or 2, as follows. The first coordinate describes which “fat row” of three 3×3 squares the chosen cell lies in; the second tells which row within that fat row it is in; the third and fourth coordinate describe the same for columns.
We use this coordinatisation, together with properties of the affine and projective spaces and other discrete structures such as Hamming codes, to give a complete description of symmetric Sudoku, a variant of Sudoku invented by Connelly and independently by Vaughan Jones.
If we regard the coordinates as lying in GF(3), we see that the rows, columns and subsquares of the Sudoku grid are affine planes. Indeed, they are defined by equations of the form
Thus, our picture above of the SET cards layout is really a picture of an isomorphism between the SET pack and the Sudoku grid, as claimed.
The five cards in positions (1,1), (1,2), (1,4), (2,1) and (4,1) determine everything. (In the Sudoku numbering these are 0000, 0001, 0010, 0100 and 1000, so the origin and the standard basis for the vector space.)
So now look at the cards in these positions, and suppose that you have chosen the numbering of attributes so that their Set coordinates are a, b, c, d, e. (Each of these is a 4-tuple, regarded as a vector in the 4-dimensional vector space.) Then the card in position with Sudoku coordinates wxyz is the one with Set coordinates a+z(b−a)+y(c−a)+x(d−a)+w(e−a) – this is the linear function which takes the values a, b, c, d, e on the five points given.
In particular, if the cards in these five positions have Set coordinates equal to their Sudoku coordinates 0000, 0001, etc., then every card in the layout will have its Set coordinates equal to its Sudoku coordinates.
There are 81 choices of the first card to go in the top right position, then 80 choices of the card to put to its right, 78 for the card beneath it (the unique card forming a set with the first two is excluded), 72 for the card in position (1,4), and 54 for the card in position (4,1). So the total number of arrangements of the pack is 81.80.78.72.54 = 1965150720, a large number but tiny by comparison with the total number 81! of orderings of the pack. The chance of producing this layout at random is vanishingly small.
Some of the arrangements are special. For example, all the cards having a particular value of a particular attribute form a 3-space. So we could arrange the cards so that the first three rows are red, the next three green, and the last three purple, for example.
It fairly often happens that the nine cards of an affine plane, laid out in a 3×3 array, form a magic square: the number of symbols in any row, column or diagonal is 6. This is because, of the 1080 lines (SETs), 729 have the property that the numbers are all different. So often we will end up with a plane in which all but three of the lines have all the numbers different. If we lay this plane out so that the main diagonal has the three cards bearing 2 symbols each, we will have a magic square as required. In fact, of the 81.80.78 = 505440 plane layouts, 27.26.54.2 = 75816 will be magic squares of this form. [A magic square must have one of the two diagonals having two symbols on its cards. Assuming it is the principal diagonal, there are 27.26 ways to choose two cards with two symbols, and 54 ways to choose a card with a different number of symbols for position (1,2): this determines the plane.]
There is another, slightly silly, kind of magic square: those in which every card carries two symbols. There are 27.26.24 = 16848 of these, giving
altogether 92664 magic squares, or 11/60 of the total number of planes.
Now expertise in computational group theory requires a wide-ranging knowledge of traditional group theory as well as a completely different take on it. There are very many groups; in some sense, the extreme cases are p-groups and (nearly) simple groups.
In 2000, the classification and counting of groups of order less than 2000 was announced by Hans Ulrich Besche, Bettina Eick and Eamonn O’Brien. There are 49 910 529 484 of them, of which almost all (approximately 49.5 billion) have order 2^{10} = 1024.
So we could give names to these groups by giving their number in the Besche–Eick–O’Brien catalogue; each group could have a 36-bit name.
Of course, unless we are much cleverer, this method cannot be extended to groups of order 2^{11} or larger powers of 2.
We can compromise. I will stick to the prime 2 here, though the same comments apply to powers of other primes. A group of order 2^{n} has a “power-commutator presentation”: it is generated by elements x_{1},…x_{n}, with defining relations giving the squares x_{i}^{2} and the commutators [x_{i},x_{j}] (for i < j) in terms of the generators. These expressions have a very special form: each is a product, over k running from i+1 to n (for the squares) or max(i,j)+1 to n (for the commutators); and the kth term is either the identity or x_{k}.
The number of expressions for the squares is thus at most (n−1)+…+1+0 = n(n-1)/2 for the squares, and (n−2)+2(n−3)+…+(n−2) = n(n−1)(n−2)/6 for the commutators, or in total (n+1)n(n−1)/6 choices.
So the number of groups of order 2^{n} is at most, roughly, 2^{(1/6)n3}. Moreover, the power-commutator presentation gives us an easily-decodable name for each group, the name having length (n+1)n(n−1)/6.
But there is a problem: not all these groups are different. The actual number of groups of order 2^{n} is asymptotically 2^{(2/27)n3}. (The lower bound was proved by Graham Higman, and the upper bound by Charles Sims.) We don’t have easily decodable names of length about (2/27)n^{3} for all groups of order n, and we don’t have an efficient way of testing isomorphism for the longer names.
This is mildly shocking, given that “most” groups of order 2^{n} have “φ-class 2″, which means that they can be described in terms of two
vector spaces over GF(2) with dimensions summing to n, together with a bilinear and a quadratic form from one space to the other, the forms related by polarisation. Here is a “simple” problem in linear algebra for which we don’t have a computationally efficient solution!
At the other end of the scale, the non-abelian finite simple groups have short names. There are a small finite number of families of “groups of Lie type”: the classical ones are parametrised by a positive integer (the rank) and a prime power (the field order), while the exceptional groups are parametrised just by a field order. The alternating groups require a single positive integer (the degree), and there are only finitely many sporadic groups.
Even if we include groups which are “nearly simple”, that is simple groups possibly extended a bit in both directions, the description is still fairly simple. This was the hypothesis behind the celebrated ATLAS of Finite Groups. The extending means that we are allowed to have a normal subgroup at the bottom which is contained in the Schur multiplier of the simple group (that is, form a central extension), and we are allowed to have a subgroup of the outer automorphism group at the top. Both the Schur multipliers and the outer automorphism groups of finite simple groups are small and well-understood.
So we might modify our requirement and say: if the group is nearly simple in this sense, give it a name; otherwise, just provide some information about it.
Probably, if a group is not “nearly simple”, we would be satisfied with some partial information about it. One thing that might be useful is a composition series for the group, telling us which simple groups go into its construction.
This tells us almost everything in the case of a nearly simple group, and almost nothing in the case of a group of prime power order. Each of the 49.5 billion groups of order 1024 has ten composition factors isomorphic to the cyclic group of order 2.
In some algorithms, such as Sylow subgroups for black box groups, rather than use traditional mathematical methods, we find a composition series first; then produce Sylow subgroups for the composition factors (these are known groups) and piece them together.
Recognising nearly simple groups as permutation groups has a long history. Indeed, my work with John Cannon addressed this question. To show that a group of permutations on n points is S_{n} or A_{n}, for example, it suffices to show that its action is primitive (which is easy), and to find a transposition or 3-cycle in the group (not so easy, but for example if we can find two elements whose supports meet in a single point then their commutator is a 3-cycle).
Recognising black-box groups is a bit more challenging. This problem was addressed for symmetric and alternating groups by Sergey Bratus and Igor Pak in 2000, and other authors have contributed subsequently. The Bratus–Pak paper (perhaps surprisingly at first view, but it makes sense on a moment’s thought) requires Goldbach’s conjecture to prove the correctness of the algorithm (the ternary Goldbach conjecture will do, and indeed the original conjecture has been verified up to values far beyond the dreams of computational group theorists), and some stronger versions of Goldbach’s conjecture are relevant to discussion of its running time. These stronger versions, concerning the number of representations of an even number as the sum of two primes, are derived from heuristics similar to that of Hardy and Littlewood for twin primes, and pose interesting challenges to number theorists.
Sad to say, a controversy has arisen over this. Igor Pak, in a blog post (I will give you the address later, but I cannot tell you what he says since the content of his blog is copyright), has alleged academic malpractice or something. Eamonn O’Brien, perhaps the only person not personally involved but having copies of all relevant emails, has posted a document giving some context not present in Pak’s original post. If you read Pak’s version, please read O’Brien’s also. The documents in question are here and here.
Recognition algorithms for other close-to-simple groups have also been given.
From some points of view, matrix groups over finite fields are perhaps the closest concrete groups to black-box groups, and the black-box algorithms are especially relevant to the matrix group recognition project, a long-term project masterminded by Charles Leedham-Green, whose aim is to find names and/or information (including composition series) for groups generated by sets of matrices. The object of the program is to produce code which can be included in the two standard computer algebra packages, Magma and GAP.
I have begun the job of putting links to full text versions of my papers into my publications list. While doing this, I observed that publishers have taken very different attitudes about how this is done. I am using the DOI mechanism wherever possible. There are two different sorts of information that publishers can provide: the actual paper (probably as a PDF file), and publication and citation information.
Here are how three academic publishers do it.