Abstract
The 2-closure \(\overline{G}\) of a permutation group G on \(\Omega\) is defined to be the largest permutation group on \(\Omega\), having the same orbits on \(\Omega \times \Omega\) as G. It is proved that ifG is supersolvable, then \(\overline{G}\) can be found in polynomial time in \(|\Omega|\). As a by-product of our technique, it is shown that the composition factors of \(\overline{G}\) are cyclic or alternating.
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References
László Babai (2018). Group, graphs, algorithms: the graph isomorphism problem. In Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. IV. Invited lectures, 3319–3336. World Sci. Publ., Hackensack, NJ
László Babai & Eugene M. Luks (1983). Canonical Labeling of Graphs. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, STOC 83, 171183. Association for Computing Machinery, New York, NY, USA
Gang Chen & Ilia Ponomarenko: Coherent Configurations. Central China Normal University Press, Wuhan (2019)
Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of finite groups. Oxford University Press, Eynsham (1985)
John D. Dixon & Brian Mortimer (1996). Permutation groups, volume 163 of Graduate Texts in Mathematics. Springer-Verlag, New York
Sergei Evdokimov & Ilia Ponomarenko: Two-closure of odd permutation group in polynomial time. Discrete Math. 235(1–3), 221–232 (2001)
Sergei Evdokimov & Ilia Ponomarenko: Circulant graphs: recognizing and isomorphism testing in polynomial time. St. Petersbg. Math. J. 15(6), 813–835 (2004)
Sergei Evdokimov & Ilia Ponomarenko: Permutation group approach to association schemes. European J. Combin. 30(6), 1456–1476 (2009)
Walter Feit (1980). Some consequences of the classification of finite simple groups. In The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), volume 37 of Proc. Sympos. Pure Math., 175–181. Amer. Math. Soc., Providence, R.I
Higman, Donald G.: Coherent configurations. I. Ordinary representation theory. Geometriae Dedicata 4(1), 1–32 (1975)
Jones, Gareth A.: Cyclic regular subgroups of primitive permutation groups. J. Group Theory 5(4), 403–407 (2002)
Martin W. Liebeck, Cheryl E. Praeger & Jan Saxl (1988). On the \(2\)-closures of finite permutation groups. J. London Math. Soc. (2)37(2), 241–252
Ponomarenko, Ilia: Graph isomorphism problem and \(2\)-closed permutation groups. Appl. Algebra Engrg. Comm. Comput. 5(1), 9–22 (1994)
Ponomarenko, Ilia: Determination of the automorphism group of a circulant association scheme in polynomial time. Zap. Nauchn. Semin. POMI 321, 251–267 (2005)
Praeger, Cheryl E., Saxl, Jan: Closures of finite primitive permutation groups. Bull. London Math. Soc. 24(3), 251–258 (1992)
Ákos Seress (2003). Permutation group algorithms, volume 152 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge
Skresanov, Saveliy V.: Counterexamples to two conjectures in the Kourovka notebook. Algebra Logic 58(3), 249–253 (2019)
Andrey V. Vasil’ev & Dmitriy (2019). The 2-closure of a \(\frac{3}{2}\)-transitive group in polynomial time. Sib. Math. J. 60(2), 279–290 (2019)
Wielandt, Helmut: Finite permutation groups. Academic Press, New York-London (1964)
Helmut Wielandt (1969). Permutation groups through invariant relations and invariant functions. The Ohio State University
Robert A. Wilson (2009). The finite simple groups, volume 251 of Graduate Texts in Mathematics. Springer-Verlag London, Ltd., London
Jing Xu, Michael Giudici, Cai Heng Li & Cheryl E. Praeger (2011). Invariant relations and Aschbacher classes of finite linear groups. Electron. J. Combin.18(1), Paper 225
Xue Yu & Jiangmin Pan: 2-closures of primitive permutation groups of holomorph type. Open Math. 17(1), 795–801 (2019)
Acknowledgements
The authors thank S. V. Skresanov and E. P. Vdovin for fruitful discussions and useful comments to the first draft of the paper, and are grateful to the anonymous referee for suggestions improving the presentation.
The work was partially supported by the RFBR Grant No. 18-01-00752, and by Mathematical Center in Akademgorodok under agreement No. 075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation.
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Ponomarenko, I., Vasil’ev, A. Two-closures of supersolvable permutation groups in polynomial time. comput. complex. 29, 5 (2020). https://doi.org/10.1007/s00037-020-00195-7
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DOI: https://doi.org/10.1007/s00037-020-00195-7