Martin W. Liebeck

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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)
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 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)
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)