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

(Extended Abstract)
  • László Babai
  • Paolo Codenotti
  • Youming Qiao
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)

Abstract

We consider the problem of testing isomorphism of groups of order n given by Cayley tables. The trivial nlogn 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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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • László Babai
    • 1
  • Paolo Codenotti
    • 2
  • Youming Qiao
    • 3
  1. 1.University of ChicagoUSA
  2. 2.University of MinnesotaUSA
  3. 3.Institute for Theoretical Computer Science, Institute for Interdisciplinary Information SciencesTsinghua UniversityChina

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