## From the archive, 2

Next up is a document which is just over a quarter-century old, a photocopy of a 26-page handwritten document entitled “On doing geometry in CAYLEY”. The background, as explained in the document, is that John Cannon’s computer algebra system CAYLEY, which had been written to facilitate computation with groups, was to be expanded to tackle a wider range of topics in discrete mathematics, and John had asked me for my views on what finite geometers would need in the new system.

At the time, I was scarcely a computer user, and certainly had no experience with CAYLEY. I had done some computing in the early 1980s, using a program in Z80 machine code on my Sinclair ZX Spectrum to prove a theorem about the infinite (a random sum-free set has positive probability of containing no even numbers). The idea was that, if pn is the probability that no even numbers have occurred as a result of the first n coin tosses, then it can be shown theoretically that pn tends to a limit p as n→∞, and that the rate of convergence can be bounded theoretically; so computing pn for sufficiently large n might allow one to conclude that p > 0 (as indeed turned out to be the case: p is about 0.218). Later I got a decent assembler for my Spectrum, which made jobs like this easier; but this was a very different kettle of fish from using CAYLEY.

Fortunately I spent a term in Sydney in the mid-1980s, and John Cannon was able to give me some flavour of what CAYLEY could do, though I didn’t then start using it. The notes are written from the perspective of a mathematician who had very little idea of what computers could do at the time or might do in future, so I was feeling my way rather cautiously.

I have no idea how much of this document went into the CAYLEY rewrite, or even who read the document. But reading it through now, I recognise several principles that Leonard Soicher and others have used very successfully. Two simple examples are the use of automorphisms to reduce the complexity of backtrack searches, and the fact that the points in a permutation representation may have come from somewhere and have their own structure.

In the intervening time, CAYLEY did indeed mutate into MAGMA; and, perhaps perversely, I started using GAP as my discrete maths computing engine. (The two systems have more similarities than differences now, in my opinion.)

Anyway, the notes surfaced. Today I was intending to finish clearing out my office, but the mathematics building is closed and the door is not functioning, so in my enforced idleness I decided to type them up. I have made no changes at all, except to correct a couple of minor “misprints” in the original handwritten text (and probably introduce some new ones). A lot of it seems old hat now, but perhaps you may find some interest here. I have put it in the “Lecture notes” section of the blog.

The project on random blocking sets referred to in Section 2 never did get done, though it did feed into the two papers with Franco Mazzocca listed below. The other three pieces of work referred to in that section found their way into print, and I did write a paper with John Cannon, in which the ideas of Section 4.3 find a use.

• P. J. Cameron and J. J. Cannon, Fast recognition of doubly transitive groups, J. Symbolic Comput. 12 (1991), 459-474.
• P. J. Cameron and P. H. Fisher, Small extended generalized quadrangles, Europ. J. Combinatorics 11 (1990), 403-413.
• P. J. Cameron, D. R. Hughes and A. Pasini, Extended generalized quadrangles, Geom Dedicata 35 (1990), 193-228.
• P. J. Cameron and F. Mazzocca, Bijections which preserve blocking sets, Geom. Dedicata 21 (1986), 219-229.
• P. J. Cameron, F. Mazzocca and R. Meshulam, Dual blocking sets in projective and affine planes, Geom. Dedicata 27 (1988), 203-207.
• P. J. Cameron and C. E. Praeger, Partitioning into Steiner systems, pp. 61-71 in Combinatorics ’88 (ed. A. Barlotti et al.), Mediterranean Press, Roma, 1992.
• P. J. Cameron and C. E. Praeger, Block-transitive t-designs, II: Large t, pp. 103-109 in Finite Geometry and Combinatorics (ed. A. Beutelspacher et al.), Cambridge Univ. Press, Cambridge, 1993.