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We present explicit upper bounds for the number and size of conjugacy classesin finite Chevalley groups and their variations. These results have been usedby many authors to study zeta functions associated to representations of finitesimple groups, random walks on Chevalley groups, the final solution to the Oreconjecture about commutators in finite simple groups and other similarproblems. In this paper, we solve a strong version of the Boston-Shalevconjecture on derangements in simple groups for most of the families ofprimitive permutation group representations of finite simple groups (theremaining cases are settled in two other papers of the authors and applicationsare given in a third).

Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with Applications to Derangements