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Nearly Linear Time Isomorphism Algorithms for Some Nonabelian Group Classes

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Computer Science – Theory and Applications (CSR 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11532))

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Abstract

The isomorphism problem for groups, when the groups are given by their Cayley tables is a well-studied problem. This problem has been studied for various restricted classes of groups. Kavitha gave a linear time isomorphism algorithm for abelian groups (JCSS 2007). Although there are isomorphism algorithms for certain nonabelian group classes, the complexities of those algorithms are usually super-linear. In this paper, we design linear and nearly linear time isomorphism algorithms for some nonabelian groups. More precisely,

  • We design a linear time algorithm to factor Hamiltonian groups. This allows us to obtain an \(\mathcal {O}(n)\) algorithm for the isomorphism problem of Hamiltonian groups, where n is the order of the groups.

  • We design a nearly linear time algorithm to find a maximal abelian factor of an input group. As a byproduct we obtain an \(\tilde{\mathcal {O}}(n)\) isomorphism for groups that can be decomposed as a direct product of a nonabelian group of bounded order and an abelian group, where n is the order of the groups.

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Notes

  1. 1.

    A group G is nilpotent class 2 if G/Z(G) is abelian.

  2. 2.

    We can compute the Sylow decomposition in \(\mathcal {O}(|G|)\) without using the result given [6], if G is Hamiltonian 2-group. Note that in a Hamiltonian 2-group order of each non-trivial element will be either 2 or 4.

References

  1. Arvind, V., Torán, J.: Solvable group isomorphism is (almost) in NP \(\cap \) coNP. ACM Trans. Comput. Theory 2(2), 4:1–4:22 (2011)

    Google Scholar 

  2. Babai, L., Codenotti, P., Qiao, Y.: Polynomial-time isomorphism test for groups with no abelian normal subgroups. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012. LNCS, vol. 7391, pp. 51–62. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31594-7_5

    Chapter  Google Scholar 

  3. Babai, L., Qiao, Y.: Polynomial-time isomorphism test for groups with abelian sylow towers. In: STACS 2012 (29th Symposium on Theoretical Aspects of Computer Science), vol. 14, pp. 453–464. LIPIcs (2012)

    Google Scholar 

  4. Boppana, R.B., Hastad, J., Zachos, S.: Does co-NP have short interactive proofs? Inf. Process. Lett. 25(2), 127–132 (1987)

    Article  MathSciNet  Google Scholar 

  5. Carmichael, R.D.: Introduction to the Theory of Groups of Finite Order. GINN and Company, Boston (1937)

    MATH  Google Scholar 

  6. Chen, L., Fu, B.: Linear and sublinear time algorithms for the basis of abelian groups. Theor. Comput. Sci. 412(32), 4110–4122 (2011)

    Article  MathSciNet  Google Scholar 

  7. Gall, F.L.: Efficient isomorphism testing for a class of group extensions. In: 26th International Symposium on Theoretical Aspects of Computer Science, STACS 2009, February 26–28, 2009, Freiburg, Germany, Proceedings, pp. 625–636 (2009)

    Google Scholar 

  8. Garzon, M., Zalcstein, Y.: On isomorphism testing of a class of \(2\)-nilpotent groups. J. Comput. Syst. Sci. 42(2), 237–248 (1991)

    Google Scholar 

  9. Grochow, J.A., Qiao, Y.: Algorithms for group isomorphism via group extensions and cohomology. SIAM J. Comput. 46(4), 1153–1216 (2017)

    Article  MathSciNet  Google Scholar 

  10. Hungerford, T.W.: Algebra. Graduate Texts in Mathematics, vol. 73. Springer, New York (1974)

    MATH  Google Scholar 

  11. Karagiorgos, G., Poulakis, D.: Efficient algorithms for the basis of finite abelian groups. Discrete Math. Algorithms Appl. 3(04), 537–552 (2011)

    Article  MathSciNet  Google Scholar 

  12. Kayal, N., Nezhmetdinov, T.: Factoring groups efficiently. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5555, pp. 585–596. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02927-1_49

    Chapter  Google Scholar 

  13. Luks, E.M.: Lectures on polynomial-time computation in groups. University of Oregon, Department of Computer and Information Science (1990)

    Google Scholar 

  14. Miller, G.L.: On the \(n^{\log n}\) isomorphism technique (a preliminary report). In: Proceedings of the Tenth Annual ACM symposium on Theory of computing, pp. 51–58. ACM (1978)

    Google Scholar 

  15. Qiao, Y.M., Sarma, J., Tang, B.S.: On isomorphism testing of groups with normal hall subgroups. J. Comput. Sci. Technol. 27(4), 687–701 (2012)

    Article  MathSciNet  Google Scholar 

  16. Rosenbaum, D.J.: Breaking the \(\cal{O}(n \log n)\) barrier for solvable-group isomorphism. In: Proceedings of The Twenty-fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1054–1073. Society for Industrial and Applied Mathematics (2013)

    Google Scholar 

  17. Rosenbaum, D.J., Wagner, F.: Beating the generator-enumeration bound for \(p\)-group isomorphism. Theor. Comput. Sci. 593, 16–25 (2015)

    Google Scholar 

  18. Savage, C.D.: An \(\cal{O}(n^2)\) algorithm for abelian group isomorphism. Computer Studies [Program], North Carolina State University (1980)

    Google Scholar 

  19. Kavitha, T.: Linear time algorithms for abelian group isomorphism and related problems. J. Comput. Syst. Sci. 73(6), 986–996 (2007)

    Article  MathSciNet  Google Scholar 

  20. Vikas, N.: An \(\cal{O}(n)\) algorithm for abelian \(p\)-group isomorphism and an \(\cal{O}(n \log n)\) algorithm for abelian group isomorphism. J. Comput. Syst. Sci. 53(1), 1–9 (1996)

    Google Scholar 

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Authors and Affiliations

  1. IIT Gandhinagar, Gandhinagar, India

    Bireswar Das & Shivdutt Sharma

Authors
  1. Bireswar Das
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  2. Shivdutt Sharma
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Correspondence to Bireswar Das or Shivdutt Sharma .

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Editors and Affiliations

  1. Novosibirsk State University, Novosibirsk, Russia

    René van Bevern

  2. Laboratoire d'Informatique Gaspard-Monge, Marne-la-Vallée, France

    Gregory Kucherov

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Das, B., Sharma, S. (2019). Nearly Linear Time Isomorphism Algorithms for Some Nonabelian Group Classes. In: van Bevern, R., Kucherov, G. (eds) Computer Science – Theory and Applications. CSR 2019. Lecture Notes in Computer Science(), vol 11532. Springer, Cham. https://doi.org/10.1007/978-3-030-19955-5_8

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  • DOI: https://doi.org/10.1007/978-3-030-19955-5_8

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  • Online ISBN: 978-3-030-19955-5

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