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Let G be a permutation group acting on a finite set Q and assume that no composition factor of G is an exceptional group of Lie type. Let p be a prime, New Monte Carlo algorit hms are presented for the construction and conj ugat ion of Sylow p-subgroups of G. The running time of the algorithms is O(STZ log’ [Gl) for some explicitly computable constant c where n is the size of Q and s is the number of generators for G. This running time is nearly linear for small-base groups. The method employs a reduction to the case of permutation represent ations of finite simple groups, where we invoke the classification of finite simple groups and design algorithms for the alternating and classical families of simple groups.

A nearly linear algorithm for Sylow subgroups in small-base groups