Journal of Computer Science and Technology

, Volume 27, Issue 4, pp 687–701 | Cite as

On Isomorphism Testing of Groups with Normal Hall Subgroups

  • You-Ming Qiao
  • Jayalal Sarma M.N.
  • Bang-Sheng Tang
Article

Abstract

A normal Hall subgroup N of a group G is a normal subgroup with its order coprime with its index. Schur-Zassenhaus theorem states that every normal Hall subgroup has a complement subgroup, that is a set of coset representatives H which also forms a subgroup of G. In this paper, we present a framework to test isomorphism of groups with at least one normal Hall subgroup, when groups are given as multiplication tables. To establish the framework, we first observe that a proof of Schur-Zassenhaus theorem is constructive, and formulate a necessary and sufficient condition for testing isomorphism in terms of the associated actions of the semidirect products, and isomorphisms of the normal parts and complement parts. We then focus on the case when the normal subgroup is abelian. Utilizing basic facts of representation theory of finite groups and a technique by Le Gall (STACS 2009), we first get an efficient isomorphism testing algorithm when the complement has bounded number of generators. For the case when the complement subgroup is elementary abelian, which does not necessarily have bounded number of generators, we obtain a polynomial time isomorphism testing algorithm by reducing to generalized code isomorphism problem, which asks whether two linear subspaces are the same up to permutation of coordinates. A solution to the latter can be obtained by a mild extension of the singly exponential (in the number of coordinates) time algorithm for code isomorphism problem developed recently by Babai et al. (SODA 2011). Enroute to obtaining the above reduction, we study the following computational problem in representation theory of finite groups: given two representations ρ and τ of a group H over \( \mathbb{Z}_p^d \), p a prime, determine if there exists an automorphism : H → H, such that the induced representation ρ𝜙 = ρ 𝜙 and τ are equivalent, in time poly(|H|, pd).

Keywords

group isomorphism normal Hall subgroup isomorphism testing problem complexity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Dehn M. Über unendliche diskontinuierliche gruppen. Mathematische Annalen, 1911, 71: 116–144.MathSciNetGoogle Scholar
  2. [2]
    Adian S. The unsolvability of certain algorithmic problems in the theory of groups. Trudy Moskov. Math. Obshch, 1957, 6: 231–298.Google Scholar
  3. [3]
    Babai L, Szemerédi E. On the complexity of matrix group problems i. In Proc. IEEE Annual Symposium on Foundations of Computer Science (FOCS), West Palm Beach, Florida, USA, October 24–26, 1984, pp.229–240.Google Scholar
  4. [4]
    Köbler J, Schöning U, Torán J. The Graph Isomorphism Problem: Its Structural Complexity. Boston: Birkhauser, 1993.MATHGoogle Scholar
  5. [5]
    Chattopadhyay A, Torán J, Wagner F. Graph isomorphism is not AC 0 reducible to group isomorphism. In Proc. Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010), Chennai, India, Dec. 15–18, 2010, pp.317–326.Google Scholar
  6. [6]
    Babai L, Luks E M. Canonical labeling of graphs. In Proc. the 15th Annual ACM Symposium on Theory of Computing (STOC), Boston, Massachusetts, USA, April 25–27, 1983, pp.171–183.Google Scholar
  7. [7]
    Miller G L. On the nlogn isomorphism technique. In Proc. the 10th Annual ACM Symposium on Theory of Computing, San Diego, California, USA, May 1–3, 1978, pp.51–58.Google Scholar
  8. [8]
    Lipton R J, Snyder L, Zalcstein Y. The complexity of word and isomorphism problems for finite groups. Technical Report, John Hopkins, 1976.Google Scholar
  9. [9]
    Savage C. An O(n 2) algorithm for abelian group isomorphism. Technical Report, North Carolina State University, 1980.Google Scholar
  10. [10]
    Vikas N. An O(n) algorithm for abelian p-group isomorphism and an O(n log n) algorithm for abelian group isomorphism. Journal of Computer and System Sciences, 1996, 53(1): 1–9.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Kavitha T. Linear time algorithms for abelian group isomorphism and related problems. Journal of Computer and System Sciences, 2007, 73(6): 986–996.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    Gall F L. Efficient isomorphism testing for a class of group extensions. In Proc. the 26th International Symposium on Theoretical Aspects of Computer Science (STACS 2009), Freiburg, Germany, February 26–28, 2009, pp.625–636.Google Scholar
  13. [13]
    Wilson J B. Decomposing p-groups via Jordan algebras. Journal of Algebra, 2009, 322(8): 2642–2679.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Wilson J B. Finding central decompositions of p-groups. Journal of Group Theory, 2009, 12(6): 813–830.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Kayal N, Nezhmetdinov T. Factoring groups efficiently. In Proc. the 36th International Colloquium on Automata, Languages and Programming (ICALP 2009), Rhodes, Greece, July 5–12, 2009, pp.585–596.Google Scholar
  16. [16]
    Wilson J B. Finding direct product decompositions in polynomial time. 2010. http://arxiv.org/pdf/1005.0548.pdf.
  17. [17]
    Taunt D R. Remarks on the isomorphism problem in theories of construction of finite groups. Mathematical Proceedings of the Cambridge Philosophical Society, 1955, 51(1): 16–24.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    Menegazzo F. The number of generators of a finite group. Irish Math. Soc. Bulletin, 2003, 50: 117–128.MathSciNetMATHGoogle Scholar
  19. [19]
    Babai L. Equivalence of linear codes. Technical Report, University of Chicago, 2010.Google Scholar
  20. [20]
    Babai L, Codenotti P, Grochow J, Qiao Y. Towards efficient algorithm for semisimple group isomorphism. In Proc. ACMSIAM Annual Symposium of Discrete Algorithms (SODA), San Francisco, California, USA, January 23–25, 2011.Google Scholar
  21. [21]
    Holt D F, Eick B, O’Brien E A. Handbook of Computational Group Theory. London: Chapman and Hall/CRC, 2005.MATHCrossRefGoogle Scholar
  22. [22]
    Rotman J J. An Introduction to the Theory of Groups (4th edition). Springer-Verlag, 1995.Google Scholar
  23. [23]
    Serre J P. Linear Representations of Finite Groups. New York: Springer-Verlag, 1977.MATHCrossRefGoogle Scholar
  24. [24]
    Rónyai L. Computing the structure of finite algebras. Journal of Symbolic Computation, 1990, 9(3): 355–373.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    Shoup V. On the deterministic complexity of factoring polynomials over finite fields. Information Processing Letters, 1990, 33(5): 261–267.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    Steel A. A new algorithm for the computation of canonical forms of matrices over fields. Journal of Symbolic Computation, 1997, 24(3–4): 409–432.MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    Seress A. Permutation Group Algorithms. Cambridge: Cambridge University Press, 2003.MATHCrossRefGoogle Scholar
  28. [28]
    Babai L. Coset intersection in moderately exponential time. Chicago Journal of Theoretical Computer Science, to appear.Google Scholar
  29. [29]
    Luks E M. Hypergraph isomorphism and structural equivalence of Boolean functions. In Proc. the 31st Annual ACM Symposium on Theory of Computing, Atlanta, Georgia, USA, May 1–4, 1999, pp.652–658.Google Scholar
  30. [30]
    Kantor W M, Luks E M, Mark P D. Sylow subgroups in parallel. Journal of Algorithms, 1999, 31(1): 132–195.MathSciNetMATHCrossRefGoogle Scholar
  31. [31]
    Babai L, Qiao Y. Polynomial-time isomorphism test for groups with abelian Sylow towers. In Proc. the 29th International Symposium on Theoretical Aspects of Computer Science, Pairs, France, Feb. 28-March 3, 2012, pp.453–464.Google Scholar
  32. [32]
    Petrank E, Roth R M. Is code equivalence easy to decide? IEEE Trans. Information Theory, 1997, 43(5): 1602–1604.MathSciNetMATHCrossRefGoogle Scholar
  33. [33]
    Buchmann J, Schmidt A. Computing the structure of a finite abelian group. Mathematics of Computation, 2005, 74(252): 2017–2026.MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    Ranum A. The group of classes of congruent matrices with application to the group of isomorphisms of any abelian group. Transactions of the American Mathematical Society, 1907, 8(1): 71–91.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC & Science Press, China 2012

Authors and Affiliations

  • You-Ming Qiao
    • 1
  • Jayalal Sarma M.N.
    • 2
  • Bang-Sheng Tang
    • 1
  1. 1.Institute for Theoretical Computer ScienceTsinghua UniversityBeijingChina
  2. 2.Department of Computer Science and EngineeringIndian Institute of Technology MadrasChennaiIndia

Personalised recommendations