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International Colloquium on Automata, Languages, and Programming

ICALP 2012: Automata, Languages, and Programming pp 51–62Cite as

Polynomial-Time Isomorphism Test for Groups with No Abelian Normal Subgroups

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7391)

Abstract

We consider the problem of testing isomorphism of groups of order n given by Cayley tables. The trivial n logn bound on the time complexity for the general case has not been improved upon over the past four decades. We demonstrate that the obstacle to efficient algorithms is the presence of abelian normal subgroups; we show this by giving a polynomial-time isomorphism test for groups without nontrivial abelian normal subgroups. This concludes a project started by the authors and J. A. Grochow (SODA 2011). Two key new ingredient are: (a) an algorithm to test permutational isomorphism of permutation groups in time, polynomial in the order and simply exponential in the degree; (b) the introduction of the “twisted code equivalence problem,” a generalization of the classical code equivalence problem by admitting a group action on the alphabet. Both of these problems are of independent interest.

Keywords

  • Group Isomorphism
  • Permutational Isomorphism
  • Code Equivalence

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Author information

Authors and Affiliations

  1. University of Chicago, USA

    László Babai

  2. University of Minnesota, USA

    Paolo Codenotti

  3. Institute for Theoretical Computer Science, Institute for Interdisciplinary Information Sciences, Tsinghua University, China

    Youming Qiao

Authors
  1. László Babai
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  2. Paolo Codenotti
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  3. Youming Qiao
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Editor information

Editors and Affiliations

  1. Department of Computer Science and Centre for Discrete Mathematics and its Applications, University of Warwick, Warwick, UK

    Artur Czumaj

  2. Max-Planck-Institut für Informatik, Saarbrücken, Germany

    Kurt Mehlhorn

  3. Computer Laboratory, University of Cambridge, UK

    Andrew Pitts

  4. ETH Zurich, Switzerland

    Roger Wattenhofer

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Babai, L., Codenotti, P., Qiao, Y. (2012). Polynomial-Time Isomorphism Test for Groups with No Abelian Normal Subgroups. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds) Automata, Languages, and Programming. ICALP 2012. Lecture Notes in Computer Science, vol 7391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31594-7_5

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  • DOI: https://doi.org/10.1007/978-3-642-31594-7_5

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