Geometry in Computer Algebra

Originally this was to be entirely on Geometry in computer algebra systems|especially CAYLEY/MAGMA and GAP (but perhaps also at least partially in other systems such as MAPLE)|to some extent just a wish list. However, at the request of John Cannon, that will be the second of two parts: I'll start with PART I. Some geometric algorithms (and applications). There are a couple of ways to phrase the question I'll discuss. The rst is this: suppose you have two permutations of 532,400 things that you know generate a primitive permutation group G, and you also know that G = PSL(3; 11). You want to know what 3 3 matrices over GF(11) produce these generators. What would you do? Thus, having a group that is simple and familiar is not the same thing as having the group in a familiar format. A better starting point involves Sylow subgroups: Input: G S n given in term of a generating set of permutations; a prime p dividing jGj. Output: A Sylow p-subgroup. (Also: Given two Sylow p-subgroups P 1 ; P 2 of G, nd g 2 G with P g 1 = P 2 .) A standard type of Sylow algorithm proceeds roughly as follows: Find g 2 G of order p, by randomly choosing elements of G until you get one of order divisible by p. Find C G (g) using backtrack. Recursively nd a Sylow p-subgroup P of C G (g) and test elements in Z(P) ? f1g to nd a nontrivial element h central in a Sylow subgroup of G. (More backtrack.) Find the set of orbits of hhi, and recursively nd a Sylow p-subgroup of the group induced on by C G (h). Pull back to a Sylow subgroup of G. Related procedures using backtrack to nd centralizers are presently employed in cayley and MAGMA (as well as GAP) in order to nd Sylow subgroups. They work well when n isn't too big. My interest is in dealing with larger n, and using an algorithm whose eeciency can be proven by some means other than trial runs. It should be noted, incidentally, that starting by nding an element of order divisible by p using random choices is now rigorously proved to be eecient in a reasonable sense IKS]; however, if the group is, say, G = PSL(3; p), with p in the hundreds of thousands, then it may be …