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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 gG 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(n 9 ). The proof makes use of the classification of finite simple groups. To solve the problems for a simple group GS n , |G|>n 8 , the ”natural” permutation representation of G is constructed in polynomial time: if GA 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 (|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 π-subgroups and finding conjugating elements for Hall π- subgroups; these algorithms are given in the Appendix.
Reviewer: P.P.Pálfy

MSC:
20D20Sylow subgroups of finite groups, Sylow properties, π-groups, π-structure
20-04Machine computation, programs (group theory)
68Q25Analysis of algorithms and problem complexity
20D05Finite simple groups and their classification