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Sylow’s theorem in polynomial time. (English) Zbl 0573.20022
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 $G\le {S}_{n}$, $|G|>{n}^{8}$, 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 $\left(|V|<{n}^{2}\right)$, 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.
Reviewer: P.P.Pálfy

##### MSC:
 20D20 Sylow subgroups of finite groups, Sylow properties, $\pi$-groups, $\pi$-structure 20-04 Machine computation, programs (group theory) 68Q25 Analysis of algorithms and problem complexity 20D05 Finite simple groups and their classification