On the covering number of small symmetric groups and some sporadic simple groups
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Luise-Charlotte Kappe
Abstract
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
Funding source: Australian Research Council
Award Identifier / Grant number: DP120101336
Funding statement: The third author acknowledges the support of the Australian Research Council Discovery Grant DP120101336 during his time spent at The University of Western Australia.
Acknowledgements
We are very thankful to Eric Borenstein, the administrator of the High Performance Computing Initiative at Florida Atlantic University, for gaining us access to KoKo and for helping us implement Gurobi on KoKo, which included finding the best parameters for optimal performance. Finally, we would like to thank Gordon Royle for giving us access to his machine at The University of Western Australia.
References
[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. Suche in Google Scholar
[2] Anderson I., Combinatorics of Finite Sets, Dover Publications, Mineola, 2002. Suche in Google Scholar
[3] Blackburn S., Sets of permutations that generate the symmetric group pairwise, J. Combin. Theory Ser. A 113 (2006), 1572–1581. 10.1016/j.jcta.2006.01.001Suche in 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. 10.1016/j.jcta.2007.07.002Suche in 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. 10.1080/00029890.1970.11992416Suche in 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. 10.1017/S0004972700036364Suche in Google Scholar
[7] Cohn J. H. E., On n-sum groups, Math. Scand. 75 (1994), 44–58. 10.7146/math.scand.a-12501Suche in 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. Suche in Google Scholar
[9]
Epstein M., Magliveras S. and Nikolova D.,
The covering numbers of
[10] Erdős P., Ko C. and Rado R., Intersection theorems for systems of finite sets, Q. J. Math. Oxford 12 (1961), 313–320. 10.1093/qmath/12.1.313Suche in Google Scholar
[11] Greco D., I gruppi che sono somma di quattro sottogruppi, Rend. Accad. Sci. Napoli 18 (1951), 74–85. Suche in Google Scholar
[12] Greco D., Su alcuni gruppi finiti che sono somma di cinque sottogruppi, Rend. Semin. Mat. Univ. Padova 22 (1953), 313–333. Suche in Google Scholar
[13] Greco D., Sui gruppi che sono somma di quattro o cinque sottogruppi, Rend. Accad. Sci. Napoli 23 (1956), 49–56. Suche in Google Scholar
[14] Haber S. and Rosenfeld A., Groups as unions of proper subgroups, Amer. Math. Monthly 66 (1959), 491–494. 10.2307/2310634Suche in Google Scholar
[15] Holmes P. E., Subgroup coverings of some sporadic groups, J. Combin. Theory Ser. A 113 (2006), 1204–1213. 10.1016/j.jcta.2005.09.006Suche in Google Scholar
[16] Holmes P. E. and Maróti A., Pairwise generating and covering sporadic simple groups, J. Algebra 324 (2010), 25–35. 10.1016/j.jalgebra.2009.10.011Suche in 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. 10.1090/conm/511/10045Suche in 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. 10.1017/CBO9780511542787.009Suche in Google Scholar
[19] Maróti A., Covering the symmetric groups with proper subgroups, J. Combin. Theory Ser. A 110 (2005), 97–111. 10.1016/j.jcta.2004.10.003Suche in Google Scholar
[20] Neumann B. H., Groups covered by permutable subsets, J. Lond Math. Soc. 29 (1954), 236–248. 10.1112/jlms/s1-29.2.236Suche in Google Scholar
[21] Scorza G., I gruppi che possone pensarsi come somma di tre sottogruppi, Boll. Unione Mat. Ital. 5 (1926), 216–218. Suche in Google Scholar
[22] Serena L., On finite covers of groups by subgroups, Advances in Group Theory 2002 (Napoli 2002), Aracne, Rome (2003), 173–190. Suche in 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. 10.1016/j.disc.2016.05.004Suche in Google Scholar
[24] Tomkinson M. J., Groups as the union of proper subgroups, Math. Scand. 81 (1997), 189–198. 10.7146/math.scand.a-12873Suche in Google Scholar
[25] The GAP Group, GAP-Groups, Algorithms, and Programming, Version 4.4.7, 2006, http://www.gap-system.org. Suche in Google Scholar
[26] Gurobi Optimizer Reference Manual, Gurobi Optimization, Inc., 2014, http://www.gurobi.com. Suche in Google Scholar
© 2016 by De Gruyter
Artikel in diesem Heft
- Frontmatter
- Computing discrete logarithms using 𝒪((log q)2) operations from {+,-,×,÷,&}
- A parallel evolutionary approach to solving systems of equations in polycyclic groups
- Authenticated commutator key agreement protocol
- On the covering number of small symmetric groups and some sporadic simple groups
- Compositions of linear functions and applications to hashing
- Hydra group doubles are not residually finite
- The status of polycyclic group-based cryptography: A survey and open problems
- On irreducible algebraic sets over linearly ordered semilattices
- A nonlinear decomposition attack
Artikel in diesem Heft
- Frontmatter
- Computing discrete logarithms using 𝒪((log q)2) operations from {+,-,×,÷,&}
- A parallel evolutionary approach to solving systems of equations in polycyclic groups
- Authenticated commutator key agreement protocol
- On the covering number of small symmetric groups and some sporadic simple groups
- Compositions of linear functions and applications to hashing
- Hydra group doubles are not residually finite
- The status of polycyclic group-based cryptography: A survey and open problems
- On irreducible algebraic sets over linearly ordered semilattices
- A nonlinear decomposition attack
