Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-13T04:32:13.391Z Has data issue: false hasContentIssue false
  • English
  • Français

Testing commutativity of a group and the power of randomization

Part of: Special aspects of infinite or finite groups Theory of computing

Published online by Cambridge University Press:  01 April 2012

Igor Pak
Igor Pak*
Affiliation:
Department of Mathematics, University of California, Los Angeles, 530 Portola Plazza 6363 Math Sciences Building Los Angeles, CA 90095-1555, USA (email: pak@math.ucla.edu)

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let G be a group generated by k elements, G=〈g1,…,gk〉, with group operations (multiplication, inversion and comparison with identity) performed by a black box. We prove that one can test whether the group G is abelian at a cost of O(k) group operations. On the other hand, we show that a deterministic approach requires Ω(k2) group operations.

MSC classification

Secondary: 20F69: Asymptotic properties of groups 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.)
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 38 - 43
Copyright
Copyright © London Mathematical Society 2012

References

[1]Arvind, V. and Mukhopadhyay, P., ‘Quantum query complexity of multilinear identity testing’, Proc. STACS 2009, Dagstuhl Seminar Proceedings (Schloss Dagstuhl, 2009) 87–98.Google Scholar
[2]Babai, L., ‘Local expansion of vertex-transitive graphs and random generation in finite groups’, Proc. STOC 1991 (ACM, New York, 1991) 164–174.Google Scholar
[3]Babai, L., ‘Randomization in group algorithms: conceptual questions’, Groups and computation II (American Mathematical Society, Providence, RI, 1997).Google Scholar
[4]Babai, L., Cooperman, G., Finkelstein, L., Luks, E. M. and Seress, Á., ‘Fast Monte Carlo algorithms for permutation groups’, J. Comput. Sys. Sci. 50 (1995) 296–308.Google Scholar
[5]Blum, M., Luby, M. and Rubinfeld, R., ‘Self-testing/correcting with applications to numerical problems’, J. Comput. Syst. Sci. 47 (1993) 549–595.CrossRefGoogle Scholar
[6]Cooperman, G. and Finkelstein, L., ‘Random algorithms for permutation groups’, CWI Quart. 5 (1992) 107–125.Google Scholar
[7]Cooperman, G. and Finkelstein, L., ‘Combinatorial tools for computational group theory’, Groups and computation I (American Mathematical Society, Providence, RI, 1993).Google Scholar
[8]Erdős, P. and Rényi, A., ‘Probabilistic methods in group theory’, J. Anal. Math. 14 (1965) 127–138.CrossRefGoogle Scholar
[9]Freivalds, R., ‘Fast probabilistic algorithms’, Mathematical foundations of computer science (Olomouc, 1979) (Springer, Berlin, 1979) 57–69.Google Scholar
[10]Frobenius, F. G., ‘Über Gruppencharaktere (in German)’, Sitz. Berl. Akad. (1896) 985–1021.Google Scholar
[11]Guralnick, R. and Wilson, J., ‘The probability of generating a finite soluble group’, Proc. Lond. Math. Soc. 81 (2000) 405–427.CrossRefGoogle Scholar
[12]Holt, D. F., Eick, B. and O’Brien, E. A., Handbook of computational group theory (Chapman & Hall/CRC, Boca Raton, FL, 2005).CrossRefGoogle Scholar
[13]Kavitha, T., ‘Linear time algorithms for abelian group isomorphism and related problems’, J. Comput. System Sci. 73 (2007) 986–996.CrossRefGoogle Scholar
[14]Luks, E. M., ‘Computing in solvable matrix groups’, Proc. FOCS 1992 (IEEE, 1992) 111–120.Google Scholar
[15]Luks, E. M., ‘Permutation groups and polynomial-time computation’, Groups and computation I (American Mathematical Society, Providence, RI, 1993).Google Scholar
[16]MacHale, D., ‘How commutative can a non-commutative group be?’, Math. Gaz. 58 (1974) 199–202.CrossRefGoogle Scholar
[17]Magniez, F. and Nayak, A., ‘Quantum complexity of testing group commutativity’, Algorithmica 48 (2007) 221–232.CrossRefGoogle Scholar
[18]Magnus, W., Karrass, A. and Solitar, D., Combinatorial group theory, 2nd edition (Dover, New York, 1976).Google Scholar
[19]Pak, I., ‘What do we know about the product replacement algorithm?’, Groups and computation, III (Columbus, OH, 1999) (de Gruyter, Berlin, 2001) 301–347.Google Scholar
[20]Rajagopalan, S. and Schulman, L. J., ‘Verifying identities’, Proc. FOCS 1996 (IEEE, 1996) 612–616.Google Scholar
[21]Santha, M., Quantum walk based search algorithms, Lecture Notes in Computer Science 4978 (Springer, Berlin) 31–46.Google Scholar
[22]Seress, A., Permutation group algorithms (Cambridge University Press, Cambridge, UK, 2003).CrossRefGoogle Scholar
[23]Shalev, A., ‘Probabilistic group theory’, Groups St. Andrews 1997 in Bath, London Mathematical Society Lecture Note Series 261 (Cambridge University Press, Cambridge, UK, 1999).Google Scholar
[24]Sims, C. C., ‘Computation with permutation groups’, Proceedings of the Second Symposium on Symbolic and Algebraic Manipulation (ACM Press, New York, 1971) 23–28.Google Scholar
[25]Tao, T. and Vu, V., Additive combinatorics (Cambridge University Press, Cambridge, UK, 2006).Google Scholar
[26]Vikas, N., ‘An O(n) algorithm for abelian p-group isomorphism and an O(nlog n) algorithm for abelian group isomorphism’, J. Comput. System Sci. 53 (1996) 1–9.CrossRefGoogle Scholar
[27]Yalçınkaya, Ş., ‘Black box groups’, Turkish J. Math. 31 (2007) 171–210.Google Scholar
[28]Zumbrägel, J., Maze, G. and Rosenthal, J., ‘Efficient recovering of operation tables of black box groups and rings’, Proceedings of ISIT 2008, Toronto, Canada (IEEE, 2008) 639–643.Google Scholar