Here is a piece of evidence which those who think that mathematics is invented rather than discovered will like.
This concerns definitions. I wrote last year about the definitions of a group, a matroid, the real numbers, and primitivity. In each case, there is a concept which is so important that it has shown up in several different contexts and can be defined in several different ways, but it is not too hard to show that the definitions are all equivalent. My story here is of a case where we are not quite so fortunate.
A permutation group G on a set X is sharply 2-transitive if, given two pairs (a1,a2) and (b1,b2) of distinct elements of X, there is a unique element of G which maps the first pair to the second. (Without the word “unique”, this would define 2-transitivity.)
Sharply 2-transitive groups arise in various contexts including model theory, universal algebras (independence algebras), and geometry (projective planes). There is no argument about the definition I have given.
However, three different mathematicians in around 1960 decided to encode a sharply 2-transitive group into an algebraic structure. Jacques Tits in 1952 used a gadget which he called a pseudofield (pseudo-corps in French); George Grätzer in 1963 used a “quasifield” (I use quotes because, in the meantime, the word “quasifield” has acquired a completely different meaning); and Helmut Karzel in 1965 used a near-domain (Fastbereich in German).
To oversimplify slightly, each of these gadgets has two operations, called addition and multiplication, and has some of the properties of a field; the elements of the group are then the affine maps of shape x → ax+b, where a is non-zero. (This is not quite correct; Grätzer used subtraction instead of addition.)
You might naively think that there should be only one natural way to do this, so the structures should all be the same. Perhaps there might be some differences; the maps might act on the left instead of the right, you might have to reverse the order of multiplication or the order of addition (or even subtraction); but basically they should be more or less the same.
With a shock, we discovered yesterday that they are not.
We asked Prover 9 whether the subtraction in a “quasifield” was a loop (in other words, the operation has left and right inverses) – this property is one of the axioms for addition in a near-domain. “No”, it said, and provided an example of a “quasifield” with three elements in which columns for the subtraction table have repeated elements. However, the maps ax−b turned out to be all the six permutations of the domain, as indeed they should be.
Subsequently Michael Kinyon found a 1972 paper by F. W. Wilke in the Bulletin of the Australian Mathematical Society showing that addition in a pseudofield need not be a loop, but that one could replace “pseudofield” by “strong pseudofield” in which addition is a loop.
We searched the web for anyone who might have looked at this and established rules for translating between the other pairs of structures – in the end without success. We asked (by email) some experienced universal algebraists whether they knew anything about this, and again drew a blank.
There must be a correspondence. Indeed, if you take an element ax+b in one structure, encode it as a permutation, and decode this as a’x+b’ in a different structure, it should be possible to express a’ and b’ in terms of a and b. Perhaps, if this were done, one could establish term equivalence, or some highbrow notion from universal algebra, of the different structures.
If you know how this is done, or if you know of a paper which does this, we would very much like to know!
While I am on the subject, I will mention a piece of fairly recent news which had escaped me; others might be interested in this as well.
Any finite sharply 2-transitive group possesses an abelian regular normal subgroup consisting of the identity and the fixed-point-free elements. (If you are thinking “Oh yes, I see how to prove that from Frobenius’ Theorem”, it is actually much easier than that; it was proved by Jordan and involves little more than simple counting.) In algebraic terms, this means that the near-domain associated with the group is a near-field.
In the case of a sharply 2-transitive group with an abelian regular normal subgroup, it is obvious, at least to a group theorist, how we should proceed. We know that any set on which a group G acts regularly can be identified with the group G, uniquely after the choice of a base point (up to an annoying glitch about left or right action.) So, if G is sharply 2-transitive on X with an abelian regular normal subgroup N, we select an element of X to be 0, then identify X with N as additive group, and the set of non-zero elements of X with the stabiliser of 0 (which acts regularly on the remaining points) as multiplicative group. The near-field axioms now more or less write themselves: addition is an abelian group, multiplication is a group with zero, and the right distributive law comes from the fact that G0 acts as automorphisms of N. In my view, it is slightly perverse to coordinatise the symmetric group of degree 3 by a structure in which the additive group is not the cyclic group of order 3.
It was an open problem for half a century whether the existence of an abelian regular normal subgroup was also true for infinite sharply 2-transitive groups and/or near-domains. This year, Rips, Segev and Tent posted a paper on the arXiv giving a counterexample (indeed, a large family of counterexamples).