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International Conference on Fundamentals of Computation Theory

FCT 1985: Fundamentals of Computation Theory pp 341–350Cite as

Probabilistic algorithms in group theory

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 199))

Abstract

A finite group G is commonly presented by a set of elements which generate G. We argue that for algorithmic purposes a considerably better presentation for a fixed group G is given by random generator set for G: a set of random elements which generate G. We bound the expected number of random elements requied to generate a given group G.

Our main results are probabilistic algorithms which take as inputs a random generator set of a fixed permutation group G \(G \subseteq S_n\). We gave O(n3 logn) expected time sequential RAM algorithms for testing membership, group inclusion and equality. Our bounds hold for any (worse case) input groups; we average only over the random generators representing the groups. Our algorithms are two orders of magnitude faster than the best previous algorithms for these group theoretic problems, which required Ω(n5) time even if given random generators.

Furthermore, we show that in the case the input group is a 2-group with a random presentation, than those group theoretic problems can be solved by a parallel RAM in O(log n)3 expected time using nO(1) processors.

This work was supported by Office of Naval Research Contract N00014-80-C-0647.

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Author information

Authors and Affiliations

  1. Aiken Computation Laboratory Division of Applied Sciences, Harvard University, Cambridge, Massachusetts

    John Reif

Authors
  1. John Reif
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Lothar Budach

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© 1985 Springer-Verlag Berlin Heidelberg

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Reif, J. (1985). Probabilistic algorithms in group theory. In: Budach, L. (eds) Fundamentals of Computation Theory. FCT 1985. Lecture Notes in Computer Science, vol 199. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0028818

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  • DOI: https://doi.org/10.1007/BFb0028818

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-15689-5

  • Online ISBN: 978-3-540-39636-9

  • eBook Packages: Springer Book Archive

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