Antiflag-transitive groups

More than 35 years ago, Bill Kantor and I published a paper classifying the finite antiflag-transitive collineation groups of classical projective and polar spaces.

This was before the Classification of Finite Simple Groups had been announced, although it was already a gleam in Daniel Gorenstein’s eye. But the roots of the paper lay earlier, in an observation by Marshall Hall. The group PSL(d,q) of d×d matrices with determinant 1 over the finite field GF(q) acts 2-transitively on the set of 1-dimensional subspaces of its vector space (that is, the points of the projective space PG(d−1,q)). Marshall Hall noticed another example of a group with this property. It happens that PSL(4,2) is isomorphic to the alternating group A8; it contains A7 as a subgroup which also happens to be doubly transitive on the 15 points of the projective space. Hall asked whether there were any other examples of this phenomenon.

I first met Bill Kantor in the spring of 1973, when I travelled across the USA on a Greyhound bus. Later in the decade he visited Oxford on sabbatical, and we worked on Hall’s question. Bill brought along the key idea for tackling it. The difficulty is that the property of 2-transitivity is not good for induction: there is no obvious way to pull out a smaller 2-transitive group.

Let G be a doubly transitive collineation group on a projective space, and let H be the stabiliser of a point P. Then H has two orbits on the points of the space; so, by Block’s lemma, it also has two orbits on the hyperplanes, which must be the hyperplanes containing P and those not containing P. Thus, G acts transitively on both flags (incident point-hyperplane pairs) and antiflags (non-incident point-hyperplane pairs). Bill saw that, in some situations, the condition of antiflag-transitivity was inherited by suitable smaller-dimensional subgroups, giving the possibility of using induction; of course, if we could find all the antiflag-transitive groups, we would in particular have an answer to Hall’s question.

So it turned out. After quite a lot of work, we managed to do this. At the same time, and with similar techniques, we could determine the antiflag-transitive subgroups of classical groups as well. (The difference in this case is that such groups preserve a form, and so there are two types of points, isotropic (on which the form vanishes) and anisotropic; following Jacques Tits, we focus on the action on isotropic points, which form the polar space associated with the group. Now the form defines a polarity on the projective space, so that the dual of a singular hyperplane is an isotropic point; thus we re-interpret antiflag transitivity as transitivity on pairs of isotropic points which are not orthogonal with respect to the form.

Our arguments involved both group theory and finite geometry (including generalized polygons and some related geometries), and I think that we did a good job. Indeed, this was where I first learned about such things as triality and the exceptional groups G2(q).

I lectured about the results in various places, though with some trepidation: being “anti-flag” doesn’t always play well!

Shortly after we did this work, I was asked by the University of London to examine the PhD thesis of Andrzej Orchel, in which a solution to Hall’s question was presented; the original examiners had declared themselves unable to determine whether the proof was correct. I had two advantages: I knew that the result was correct, and I knew how the proof should go. Orchel’s strategy was also to attack the problem by looking at antiflag-transitive groups. If I recall correctly, he called these groups T-groups; he did indeed answer Hall’s question, but stopped short of a complete classification of T-groups.

Shortly afterwards, it emerged that there were two mistakes in our paper, which led to the omission of some antiflag-transitive groups. I am sad about this because one of the omitted cases is so beautiful.

Consider the 8-dimensional space over the binary field GF(2) consisting of all words of length 9 having even weight. This form carries a bilinear form, the usual dot product: because of the even weight condition, any vector has norm zero, so the form is symplectic. There is also a quadratic form, which polarises to the symplectic form, given by taking Q(v) to be half the weight of v (mod 2), that is, the singular vectors are those with weight 4 or 8. The symmetric group S9 acts on the vector space preserving these forms, and the alternating group A9 is contained in the simple orthogonal group.

But this group admits a triality automorphism, which geometrically maps the sets of points and two types of maximal singular subspaces on the quadric in a 3-cycle. The image of the A9 just constructed has an orbit of length 9 on maximal subspaces (forming a spread on the quadric) and acts transitively (and indeed antiflag-transitively) on the points of the quadric. Somehow we had omitted this lovely example!

A simple fix puts the argument right; I wrote this up and sent it as a corrigendum to the journal, but they never published it, and I was too busy at the time to chase it up. About half a page of argument suffices. If I find my correction I will post it, but I fear it may have gone when I cleared out my office.

The other error is in some ways more serious, though very easily fixed; it leads to the omission of an infinite family of antiflag-transitive examples. See below.

Many years later, somebody queried the correctness of the paper. I acknowledged that there was a mistake, and that we had fixed it, but that I could not at the time remember how the fix went. So I decided to write up from scratch a complete account of our proof. I made a start, but didn’t get very far before I found myself (as so often happens) floundering in lots of urgent jobs, and abandoned it.

Now Bill Kantor wrote to me saying that search engines have found the incomplete draft and have contributed to a feeling that there is something wrong with the proof. So, at his urging, here is the second correction.

On page 401, line 2, we omitted the possibility
q = 4, δ = e = 2
leading to the antiflag-transitive example
ΓL(m,16) inside ΓL(2m,4).
This does not influence any other part of the paper.


About Peter Cameron

I count all the things that need to be counted.
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