Finite primitive permutation groups of rank 3 have been the subject of much study in the past twenty years, leading on the one hand to interesting new groups (for instance, some sporadic groups) and on the other to new techniques in the theory of permutation groups. It is readily seen that if G is a primitive rank 3 permutation group of finite degree n then… (More)
We survey recent progress, made using probabilistic methods , on several conjectures concerning nite groups. 1 Random generation In recent years probabilistic methods have proved useful in the solution of several diicult problems concerning nite groups; these involve conjectures on nite simple groups and on nite permutation groups. In some cases the… (More)
A classification is given of primitive K 0-categorical structures which are smoothly approximated by a chain of finite homogeneous substructures. The proof uses the classification of finite simple groups and some representation theory. The main theorems give information about a class of structures more general than the X 0-categorical, co-stable structures… (More)
We present a black-box polynomial-time Monte Carlo algorithm which, given as input a quasi-simple group of Lie type, finds its characteristic.
Let G be a finite simple group and let S be a normal subset of G. We determine the diameter of the Cayley graph Γ(G, S) associated with G and S, up to a multiplicative constant. Many applications follow. For example, we deduce that there is a constant c such that every element of G is a product of c involutions (and we generalize this to elements of… (More)
We extend a result of E. Hrushovski and A. Pillay as follows. Let G be a finite subgroup of GL(n, F) where F is a field of characteristic p such that p is sufficiently large compared to n. Assume that G is generated by p-elements. Then G is a product of 25 of its Sylow p-subgroups. If G is a simple group of Lie type in characteristic p, the analogous result… (More)
We study finite subgroups of exceptional groups of Lie type, in particular maximal subgroups. Reduction theorems allow us to concentrate on almost simple subgroups, the main case being those with socle X(q) of Lie type in the natural characteristic. Our approach is to show that for sufficiently large q (usually q > 9 suffices), X(q) is contained in a… (More)