Andrzej Orchel

This month’s LMS Newsletter carries a brief item:

Dr Andrzej W. Orchel, who was elected a member of the London Mathematical Society on 22 November 1969, died on 22 January 2012, aged 65.

It is not just the coincidence that he died a day before my 65th birthday that led me to write this. In November 1969 I had just begun studying at Oxford, and there was no question of my joining the London Mathematical Society then.

One of the best pieces of work I did in the 1970s, jointly with Bill Kantor, was to resolve a problem of Marshall Hall: show that the only subgroup of PΓL(n,q) which acts doubly transitively on the points of the projective space and does not contain the “little projective group” PSL(n,q) is the alternating group A7, a subgroup of PGL(4,2) (which happens to be isomorphic to A8.

Bill’s insight was that where previous attempts had failed because there is nothing to get a grip on, the right observation was that a 2-transitive group on the points of a projective space is antiflag-transitive on the space, that is, transitive on non-incident point-hyperplane pairs. This allowed a limited amount of induction; perhaps we could classify the antiflag-transitive groups, and get the proof of Hall’s conjecture as a corollary.

Indeed, on looking further, we found that in fact antiflag-transitive groups which are not 2-transitive preserve very nice geometric structures, of which the most important cases are generalized polygons; we were able to deploy results about these, such as the Feit–Higman Theorem, in our argument.

We published the result in the Journal of Algebra. We noticed later that we had left a single group out of our conclusions, namely the alternating group A9; we were easily able to fix the argument to cover this, and wrote a very short corrigendum for the journal (which I think was never published).

In 1979, when the LMS awarded me one of the inaugural Junior Whitehead Prizes, this paper was one of my works which was cited. In fact, we had by the skin of our teeth saved this very nice problem from being demolished by the Classification of Finite Simple Groups. The CFSG, announced in 1980 (though not proved for another quarter-century), together with results already known, gives a complete list of all doubly-transitive groups, and the answer to Hall’s question can be read off.

But the affair had a curious spin-off, which happened after we had proved the theorem but before it was published. (I am speaking from memory in the next bit, since all my records from that time have been thrown away.)

I was invited by the University of London to be the external examiner of a PhD thesis which also established the truth of Hall’s conjecture. It turned out that a previous examiner, who had been unable to understand the thesis, had withdrawn, and they called me in to replace him.

The candidate was Andrzej W. Orchel.

I struggled with the thesis, and had some sympathy for the examiner who had withdrawn. But I had a secret weapon; since I had a proof of my own of the theorem, I knew how the argument had to go, and so at points where the writing wasn’t clear I was able to find my way. If my memory is correct, Orchel also used antiflag-transitive groups – I think he called them T-groups – but he did not classify all such groups, merely learned enough about them to classify the doubly transitive ones.

So we examined Orchel and passed him. (I am not sure who the internal examiner was; possibly Fred Piper. I am also not sure who Orchel’s supervisor was; possibly Otto Wagner.)

I think he was hoping to have an academic career, but I’m afraid I lost touch with him. However, Google Books lists a book he published in 1979, entitled Finite Groups and Associated Geometric Structures, and gives his address as Queen Elizabeth College (one of the parts of the University of London which later disappeared in the re-structuring, swallowed up by King’s College).

Googling his name gives a number of links, almost all in Polish (the exceptions are the book and the LMS Newsletter). I haven’t had time to try to find out more.

But I got quite a jolt when I saw the notice.

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About Peter Cameron

I count all the things that need to be counted.
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