L. Babai, A. J. Goodman, W. M. Kantor, E. M. Luks and P. P. Palfy, Short presentations for finite groups. J. Algebra 194 (1997) 79--112.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....system. A presentation for A M can either be computed directly from a representation for A M or by first computing a presentation for A and adding words for generators of M . For permutation groups a presentation can be computed either via a stabilizer chain [4] or from a composition series [1, 14]. The latter usually yields a better presentation and can also be modified to yield a presentation for A M without the need to compute a permutation representation of this group first. The complements parametrized by Z 1 are all normal in A but not necessarily normal in G. We therefore have to ....

Babai, L., Goodman, A. J., Kantor, W. M., Luks, E., and P alfy, P. P. Short presentations for finite groups. J. Algebra 194 (1997), 97--112.

....solved in NP coAM. In the same paper a conjecture is framed for nite groups called the Short Presentation Conjecture, then it is shown that under this conjecture the subgroup membership problem can be solved in NP coNP. There is a lot of evidence in favor of the conjecture, in particular in [BGKLP97] it is shown that if the conjecture holds for nite simple groups then it holds for all nite groups. Further, the conjecture has been veri ed for all nite simple groups except for three families of groups namely, the unitary groups PSU(3; q) the Suzuki groups Sz(q) and the Ree groups R(q) see ....

....it is shown that if the conjecture holds for nite simple groups then it holds for all nite groups. Further, the conjecture has been veri ed for all nite simple groups except for three families of groups namely, the unitary groups PSU(3; q) the Suzuki groups Sz(q) and the Ree groups R(q) see [BGKLP97]) It follows from the results in that paper that if none of these exceptional families of groups occur as factor groups of composition series of subgroups of a group G, then the subgroup membership problem for G can be solved in NP coNP. In this article we show that there are in nitely many ....

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Babai, L.; Goodman, A., J.; Kantor, W., M.; Luks, E., M., and Palfy, P., P.; Short Presentations for Finite Groups, J. Algebra, (194), no. 1, 79-112, 1997.

....classical groups it is necessary to exclude the four dimensional symplectic groups. ffl Almost all pairs of elements of the symmetric group give rise to Cayley graphs with small diameter (Babai and Hetyei [5] and at least some give short presentations (Liebeck and Shalev [70] See also [6, 4, 71]. 5 Extremal properties A number of metrics have been considered for permutations, including distance in the Cayley graph (with respect to the generating set consisting of transpositions) the Hamming and Lee metrics, l 1 , l 2 and l 1 metrics, and commutation distance. Extremal problems ....

L. Babai, A. J. Goodman, W. M. Kantor, E. M. Luks and P. P. P'alfy, Short presentations for finite groups, J. Algebra 194 (1997), 79--112.

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L. Babai, A. J. Goodman, W. M. Kantor, E. M. Luks and P. P. Palfy, Short presentations for finite groups. J. Algebra 194 (1997) 79--112.

....a permutation g given as a product of generators, verifying that g = 1 involves the lengths of presentations 5 . For all simple groups except, perhaps, PSU(3; q) 2 B 2 (q) and 2 G 2 (q) there is a presentation of length O(log c jGj) using c = 2 and in most cases even c = 1; the proof in [BGKLP] uses simple tricks to adapt the usual Curtis Steinberg Tits presentations for these groups. It is perhaps surprising that the case of the very familiar groups PSU(3; q) has remained open for almost 10 years. Short presentations have the following nonalgorithmic consequence needed in the proof of ....

L. Babai, A. J. Goodman, W. M. Kantor, E. M. Luks and P. P. P'alfy, Short presentations for finite groups. J. Algebra 194 (1997) 79--112.

....also constructed a composition series for G. In this respect, the algorithm resembles the parallel handling of permutation groups [BLS1] and the current fastest deterministic algorithms for computing strong generating sets [BLS2, BLS3] The second theorem is a constructive version of a result from [BGKLP] about short presentations of groups. Theorem 1.3 There is a nearly linear Las Vegas algorithm which, when given a permutation group G satisfying the composition factor restriction of Theorem 1.2, computes a presentation of length O(log 3 jGj) for G. Using the terminology of this paper, the ....

....on the groups of Lie rank 2 other than 2 F 4 (q) Possibly the biggest obstacle is condition (ii) of Definition 1. 1 in the case of rank 1 groups: finding O(log c jGj) length presentations for PSU(3; q) 2 B 2 (q) and 2 G 2 (q) has been a very annoying open problem for several years (cf. [BGKLP]) 2 The proofs 2.1 Bases, strong generating sets, and Schreier trees Fundamental data structures for computing with permutation groups were introduced by Sims in [Si1, Si2] A base for a permutation group G Sym( Omega Gamma of degree n is a sequence B = fi 1 ; fi M ) of points ....

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L. Babai, A. J. Goodman, W. M. Kantor, E. M. Luks and P. P. P'alfy, Short presentations for finite groups, J. Algebra 194 (1997), 79--112. 12 W. M. KANTOR AND ' A. SERESS

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L. Babai, A. J. Goodman, W. M. Kantor, E.M. Luks, and P. P. P alfy, Short presentations for finite groups, J. Algebra 194 (1997), 97--112.

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