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A classification is given of primitive K0-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 X0-categorical, co-stable structures… (More)

(iii) G is an ajfine group, that is, the socle of G is a vector space V, where V s= {ZpY for some prime p and n=p ; moreover, if Go is the stabilizer of the zero vector in V then G = VG0, Go is an irreducible subgroup of GLd(p), and GQ has exactly two orbits on the non-zero vectors of V. The rank 3 groups under (i) are given by the classification of the… (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)

- Robert M. Guralnick, Martin W. Liebeck, Dugald Macpherson, Gary M. Seitz
- 1997

We prove that the number of conjugacy classes of maximal subgroups of bounded order in a finite group of Lie type of bounded rank is bounded. For exceptional groups this solves a longstanding open problem. The proof uses, among other tools, some methods from Geometric Invariant Theory. Using this result we provide a sharp bound for the total number of… (More)

In this paper we prove some mainly asymptotic results concerning the irreducible character degrees of .nite groups of Lie type. Applications are given to the study of the mixing time of random walks on these groups, with certain conjugacy classes as generating sets. In various situations we show that the mixing time is 2; this seems to be the .rst… (More)

- Martin W. Liebeck, Gary M. Seitz
- 2002

Let G be a simple algebraic group of exceptional type G2, F4, E6, E7 or E8 over an algebraically closed field K of characteristic p. The analysis of maximal subgroups of exceptional groups has a history stretching back to the fundamental work of Dynkin [3], who determined the maximal connected subgroups of G in the case where K has characteristic zero. The… (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 thatG is generated by p-elements. ThenG 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)

In recent years probabilistic methods have proved useful in the solution of several problems concerning finite groups, mainly involving simple groups and permutation groups. In some cases the probabilistic nature of the problem is apparent from its very formulation (see [KL], [GKS], [LiSh1]); but in other cases the use of probability, or counting, is not… (More)