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Groups Complexity Cryptology

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On the covering number of small symmetric groups and some sporadic simple groups

Luise-Charlotte Kappe
  • Department of Mathematical Sciences, State University of New York at Binghamton, Binghamton, NY 13902-6000, United States of America
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/ Daniela Nikolova-Popova
  • Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431, United States of America
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/ Eric SwartzORCID iD: http://orcid.org/0000-0002-1590-1595
Published Online: 2016-10-12 | DOI: https://doi.org/10.1515/gcc-2016-0010


A set of proper subgroups is a cover for a group if its union is the whole group. The minimal number of subgroups needed to cover G is called the covering number of G, denoted by σ(G). Determining σ(G) is an open problem for many nonsolvable groups. For symmetric groups Sn, Maróti determined σ(Sn) for odd n with the exception of n=9 and gave estimates for n even. In this paper we determine σ(Sn) for n=8,9,10,12. In addition we find the covering number for the Mathieu group M12 and improve an estimate given by Holmes for the Janko group J1.

Keywords: Symmetric groups; sporadic simple groups; finite union of proper subgroups; minimal number of subgroups

MSC 2010: 20D06; 20D60; 20D08; 20-04; 20F99


  • [1]

    Abdollahi A., Ashraf F. and Shaker S. M., The symmetric group of degree six can be covered by 13 and no fewer proper subgroups, Bull. Malays. Math. Sci. Soc. 30 (2007), 57–58. Google Scholar

  • [2]

    Anderson I., Combinatorics of Finite Sets, Dover Publications, Mineola, 2002. Google Scholar

  • [3]

    Blackburn S., Sets of permutations that generate the symmetric group pairwise, J. Combin. Theory Ser. A 113 (2006), 1572–1581. Google Scholar

  • [4]

    Britnell J. R., Evseev A., Guralnick R. M., Holmes P. E. and Maróti A., Sets of elements that pairwise generate a linear group, J. Combin. Theory Ser. A 115 (2008), 442–465. Google Scholar

  • [5]

    Bruckheimer M., Bryan A. C. and Muir A., Groups which are the union of three subgroups, Amer. Math. Monthly 77 (1970), 52–57. Google Scholar

  • [6]

    Bryce R. A., Fedri V. and Serena L., Subgroup coverings of some linear groups, Bull. Aust. Math. Soc. 60 (1999), 239–244. Google Scholar

  • [7]

    Cohn J. H. E., On n-sum groups, Math. Scand. 75 (1994), 44–58. Google Scholar

  • [8]

    Conway J. H., Curtis R. T., Norton S. P., Parker R. A. and Wilson R. A., Atlas of Finite Groups, Oxford University Press, Oxford, 2005. Google Scholar

  • [9]

    Epstein M., Magliveras S. and Nikolova D., The covering numbers of A9 and A11, J. Combin. Math. Combin. Comput., to appear. Google Scholar

  • [10]

    Erdős P., Ko C. and Rado R., Intersection theorems for systems of finite sets, Q. J. Math. Oxford 12 (1961), 313–320. Google Scholar

  • [11]

    Greco D., I gruppi che sono somma di quattro sottogruppi, Rend. Accad. Sci. Napoli 18 (1951), 74–85. Google Scholar

  • [12]

    Greco D., Su alcuni gruppi finiti che sono somma di cinque sottogruppi, Rend. Semin. Mat. Univ. Padova 22 (1953), 313–333. Google Scholar

  • [13]

    Greco D., Sui gruppi che sono somma di quattro o cinque sottogruppi, Rend. Accad. Sci. Napoli 23 (1956), 49–56. Google Scholar

  • [14]

    Haber S. and Rosenfeld A., Groups as unions of proper subgroups, Amer. Math. Monthly 66 (1959), 491–494. Google Scholar

  • [15]

    Holmes P. E., Subgroup coverings of some sporadic groups, J. Combin. Theory Ser. A 113 (2006), 1204–1213. Google Scholar

  • [16]

    Holmes P. E. and Maróti A., Pairwise generating and covering sporadic simple groups, J. Algebra 324 (2010), 25–35. Google Scholar

  • [17]

    Kappe L.-C. and Redden J. L., On the covering number of small alternating groups, Computational Group Theory and the Theory of Groups. II, Contemp. Math. 511, American Mathematical Society, Providence (2010), 93–107. Google Scholar

  • [18]

    Lucido M. S., On the covers of finite groups, Groups St. Andrews 2001 in Oxford, London Math. Soc. Lecture Note Ser. 305, Cambridge University Press, Cambridge (2003), 395–399. Google Scholar

  • [19]

    Maróti A., Covering the symmetric groups with proper subgroups, J. Combin. Theory Ser. A 110 (2005), 97–111. Google Scholar

  • [20]

    Neumann B. H., Groups covered by permutable subsets, J. Lond Math. Soc. 29 (1954), 236–248. Google Scholar

  • [21]

    Scorza G., I gruppi che possone pensarsi come somma di tre sottogruppi, Boll. Unione Mat. Ital. 5 (1926), 216–218. Google Scholar

  • [22]

    Serena L., On finite covers of groups by subgroups, Advances in Group Theory 2002 (Napoli 2002), Aracne, Rome (2003), 173–190. Google Scholar

  • [23]

    Swartz E., On the covering number of symmetric groups having degree divisible by six, Discrete Math. 339 (2016), no. 11, 2593–2604. Google Scholar

  • [24]

    Tomkinson M. J., Groups as the union of proper subgroups, Math. Scand. 81 (1997), 189–198. Google Scholar

  • [25]

    The GAP Group, GAP-Groups, Algorithms, and Programming, Version 4.4.7, 2006, http://www.gap-system.org.

  • [26]

    Gurobi Optimizer Reference Manual, Gurobi Optimization, Inc., 2014, http://www.gurobi.com.

About the article

Received: 2016-02-17

Published Online: 2016-10-12

Published in Print: 2016-11-01

Funding Source: Australian Research Council

Award identifier / Grant number: DP120101336

The third author acknowledges the support of the Australian Research Council Discovery Grant DP120101336 during his time spent at The University of Western Australia.

Citation Information: Groups Complexity Cryptology, Volume 8, Issue 2, Pages 135–154, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc-2016-0010.

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