The main theorem of the paper states that there are polynomial-time algorithms which, when given a subgroup G of the symmetric group

${S}_{n}$ and a prime p, solve the following problems: (i) given a p-subgroup P of G, find a Sylow p-subgroup of G containing P; and (ii) given Sylow p- subgroups

${P}_{1}$,

${P}_{2}$ of G, find

$g\in G$ conjugating

${P}_{1}$ to

${P}_{2}$. (G and its subgroups are specified in terms of generating permutations.) The result is mainly of theoretical interest as the running time of the algorithms is

$O\left({n}^{9}\right)$. The proof makes use of the classification of finite simple groups. To solve the problems for a simple group

$G\le {S}_{n}$,

$\left|G\right|>{n}^{8}$, the ”natural” permutation representation of G is constructed in polynomial time: if

$G\simeq {A}_{m}$ then an m-element set and the action of G on it; if G is isomorphic to a classical group then a vector space V

$\left(\right|V|<{n}^{2})$, a form on V if G is symplectic, orthogonal, or unitary, and the action of G on the set of 1-spaces of V. Having a solution for simple groups, the algorithms consist of several reductional procedures. For solvable groups the algorithms can be simplified and extended to finding Hall

$\pi $-subgroups and finding conjugating elements for Hall

$\pi $- subgroups; these algorithms are given in the Appendix.