W.M. Kantor, Sylow'stheorem in polynomial time, J. Comp. Syst. Sci. 30 (1985), 359-394.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in Section 5. Acknowledgment. The authors wish to thank the referees for their valuable comments. 2. Preliminaries We recall portions of the polynomial time library for permutation groups. For more details, we refer to the survey article [19] We begin with the following results (cf. 7] [11], 12] 17] 29] Theorem 2.1. Given G Sym( in polynomial time one can solve the following problems. i) Given , find the orbit of under G and test transitivity of G. ii) Test the primitivity of G and, if not, find a non trivial block system. iii) Given x 2 Sym( test ....

.... would have no such extension. Theorem 2.3 (Kantor Luks) Problems (iv) v) vi) vii) and (viii) of Theorem 2.1 and problems (ii) and (iii) of Theorem 2.2 remain in polynomial time if G = L=K for K C L Sym( Remark. In general, the Sylow algorithms (Theorem 2. 1 (v) [11], 12] strongly depend on the Classification of the Finite Simple Groups. In turn, these algorithms are critical to the quotient group versions of Theorem 2.2 (ii) and (iii) However, the Classification is not needed for any of these results when G 2 d . In the spirit of the quotient group ....

W. M. KANTOR, Sylow's theorem in polynomial time, J. Comput. System Sci. 30 (1985), 359--394.

....[Luk87] The analysis of Luks algorithm uses many deep permutation group theoretic results along with a detailed knowledge of the classification of finite simple groups. The problem of computing a generator set of a Sylow p subgroup for prime p, was shown to be in polynomial time by W. Kantor [Kan85] A library of polynomial time algorithms for a long list of problems over permutation groups (even for factor groups) is given in [KL90] In [BLS87] Babai et al. have shown that membership testing, order computation, computing the center and the composition series can be performed in NC. These ....

W. Kantor. Sylow's theorem in polynomial time. Journal of Computer and System Sciences, 30:359--394, 1985.

....that kxk ckyk for all y 2 M Gamma n 0 o . Reference: Gat84] Remarks: IntegerGCD, Problem B.1.1, is NC 1 reducible to SV2 [Gat84] B.1.8 Sylow Subgroups (SylowSub) Given: A group G. Problem: Find the Sylow subgroups of G. Reference: BSL87] Remarks: The problem is known to be in P [Kan85], however, the NC question is open even for solvable groups [BSL87] For a permutation group G, testing membership in G, finding the order of G, finding the center of G, and finding a composition series of G are all known to be in NC [BSL87] Babai, Seres, and Luks present several other open ....

W. M. Kantor. Sylow's theorem in polynomial time. Journal of Computer and System Sciences, 30(3):359--394, 1985.

....cases up into the hundred thousands. It is particularly interesting to note that some of these new methods for permutation groups, which have now become very practical, too, first were brought in through rather theoretical discussions of the complexity of permutation group algorithms. Neu86] Kan85] KT88] BLS88] BCF ar] BCFS91] BS92] to mention just a small selection of many papers on this subject. A broad variety of methods, many interactive, are available for working with representations and characters [NPP84] LP91] A long neglected, but now very rapidly growing branch of ....

W. Kantor, Sylow's theorem in polynomial time, J. Comput. Syst. Sci. 30 (1985), 359--394.

....x such that kxk ckyk for all y 2 M 0 n 0 o . Reference: Gat84] Remarks: IntegerGCD, Problem B.1.1, is NC 1 reducible to SV2 [Gat84] B.1.8 Sylow Subgroups (SylowSub) Given: A group G. Problem: Find the Sylow subgroups of G. Reference: BSL87] Remarks: The problem is known to be in P [Kan85], however, the NC question is open even for solvable groups [BSL87] For a permutation group G, testing membership in G, finding the order of G, finding the center of G, and finding a composition series of G are all known to be in NC [BSL87] Babai, Seres, and Luks present several other open ....

W. M. Kantor. Sylow's theorem in polynomial time. Journal of Computer and System Sciences, 30(3):359--394, 1985.

....the correctness of the answer. A second generation of algorithms uses divide and conquer techniques by utilizing the imprimitivity block structure of the input group, thereby reducing the problems to primitive groups. Besides [64] we mention [65] 81] for computing a composition series, [49], 50] for Sylow subgroups, and the asymptotically fastest deterministic SGS constructions [9] 8] Although, in [8] an in depth structural examination of G is necessary before jGj is obtained, namely, the computation of a composition series, the algorithm runs almost a factor n 2 faster than ....

....computation of a composition series, the algorithm runs almost a factor n 2 faster than the asymptotically fastest versions of Sims original method. The required group theoretic arsenal for these algorithms include consequences of the classification of finite simple groups and, in the case of [49], 50] detailed knowledge of the classical simple groups. We mention, however, that a composition series can be computed without using the simple group classification [12] The running time of most algorithms for G = hSi S n does not only depend on the input length jSjn, but also on the order ....

William M. Kantor. Sylow's theorem in polynomial time. J. Comp. Syst. Sci., 30:359--394, 1985.

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W. M. Kantor, Sylow's theorem in polynomial time, J. Comp. Syst. Sci. 30 (1985), 359--394.

....to the graph isomorphism problem (ISO) for early work ( Ba1] FHL] Lu1] Mi1] Mi2] BL] used groups to put significant instances of ISO into polynomial time. Ensuing studies resulted in algorithms for deciphering the basic building blocks of the group ( BKL] Lu2] Ne] KT] Ka1] [Ka2], Ka3] BLS1] making available constructive versions of standard theoretical tools. One essential ingredient has, to a great extent, been lacking. The facility to deal with quotient groups (equivalently, homomorphic images of groups) is a central methodology of group theory, but there has not ....

....of the set) Lu2] and so computable by the most basic algorithm in [FHL] Such a concrete interpretation is not available for quotient groups. To show that the center of a quotient group can be computed efficiently, we make essential use of the Sylow structure made available via procedures in [Ka2], Ka3] procedures that are dependent upon the ( 15000 page) classification of finite simple groups. Nevertheless, we emphasize that our new methods do not require a deep knowledge of this classification or of other algebraic theory of great depth. The procedures that we cite (e.g. from [Lu2] ....

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W.M. Kantor, Sylow's theorem in polynomial time, J. Comp. Syst. Sci. 30 (1985) 359-394.

....is, say, G = PSL(3; p) with p in the hundreds of thousands, then it may be necessary to make almost p choices, and hence take unacceptably long to find such an element in this manner. There is an alternative approach, originally developed in a purely theoretical (polynomial time) framework [Ka2]. This involves a reduction to two problems: ffl Find a Sylow p subgroup of a simple group G. ffl Given two Sylow p subgroups of a simple group G, find an element of g conjugating one to the other. N.B. CAYLEY, MAGMA and GAP can find elements that conjugate a given subgroup to any known ....

....this is not very good for moderate sized n. It is important to understand that the reduction makes essential use of the ability to conjugate Sylow subgroups of simple groups: this leads to an effective form of the Frattini argument. I won t discuss this reduction. There are a few versions of it [Ka2, Ka3, KLM]. The more times it is reworked the slicker it gets. There is a parallel version of the reduction (for theoretical purposes at this stage [KLM] There is a nearly linear time 2 version being developed (by Prabhav Morje, a PhD student of Akos Seress, and Morje will be programming several parts ....

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W. M. Kantor, Sylow's theorem in polynomial time. J. Comp. Syst. Sci. 30 (1985) 359--394.

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W.M. Kantor, Sylow'stheorem in polynomial time, J. Comp. Syst. Sci. 30 (1985), 359-394.

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W. M. Kantor. Sylow's theorem in polynomial time. J. Comp. Syst. Sci., 30:359--394, 1985.

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W.M. Kantor, Sylow's theorem in polynomial time, J. Comp. Syst. Sci. 30 (1985) 359-394.

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W. M. Kantor, Sylow's theorem in polynomial time, J. Comut. System Sci. 30 (1985), 359-- 394.

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