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The covering number of $M_{24}$

Year 2016, Volume: 3 Issue: 3, 155 - 158, 09.08.2016
https://doi.org/10.13069/jacodesmath.90728

Abstract

A  finite cover $\mathcal{C}$ of a group $G$ is a finite collection of proper subgroups of $G$ such that $G$ is equal to the union of all of the members of $\mathcal{C}$. Such a cover is called {\em minimal} if it has the smallest cardinality among all finite covers of $G$. The  covering number of $G$, denoted by $\sigma(G)$, is the number of subgroups in a minimal cover of $G$. In this paper the covering number of the Mathieu group $M_{24}$ is shown to be 3336.

References

  • W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24(3-4) (1997) 235–265.
  • R. A. Bryce, V. Fedri, L. Serena, Subgroup coverings of some linear groups, Bull. Austral. Math. Soc. 60(2) (1999) 227–238.
  • J. H. E. Cohn, On n-sum groups, Math. Scand. 75(1) (1994) 44–58.
  • J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
  • M. Epstein, S.S. Magliveras, D. Nikolova-Popova, The covering numbers of $A_9$ and $A_11$, to appear in the J. Combin. Math. Combin. Comput.
  • P. E. Holmes, Subgroup coverings of some sporadic groups, J. Combin. Theory Ser. A 113(6) (2006) 1204–1213.
  • L. C. Kappe, J. L. Redden, On the covering number of small alternating groups, Contemp. Math. 511 (2010) 109–125.
  • L. C. Kappe, D. Nikolova-Popova, E. Swartz, On the covering number of small symmetric groups and some sporadic simple groups, arXiv:1409.2292v1 [math.GR].
  • M. S. Lucido, On the covers of finite groups, in: C. M. Campbell, E. F. Robertson, G. C. Smith (Eds), Groups St. Andrews 2001, in Oxford, vol II, in : London Math. Soc. Lecture Note Ser. 305, 2003, 395–399.
  • A. Maróti, Covering the symmetric groups with proper subgroups, J. Combin. Theory Ser. A 110(1) (2005) 97–111.
  • M. J. Tomkinson, Groups as the union of proper subgroups, Math. Scand. 81(2) (1997) 191–198.
Year 2016, Volume: 3 Issue: 3, 155 - 158, 09.08.2016
https://doi.org/10.13069/jacodesmath.90728

Abstract

References

  • W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24(3-4) (1997) 235–265.
  • R. A. Bryce, V. Fedri, L. Serena, Subgroup coverings of some linear groups, Bull. Austral. Math. Soc. 60(2) (1999) 227–238.
  • J. H. E. Cohn, On n-sum groups, Math. Scand. 75(1) (1994) 44–58.
  • J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
  • M. Epstein, S.S. Magliveras, D. Nikolova-Popova, The covering numbers of $A_9$ and $A_11$, to appear in the J. Combin. Math. Combin. Comput.
  • P. E. Holmes, Subgroup coverings of some sporadic groups, J. Combin. Theory Ser. A 113(6) (2006) 1204–1213.
  • L. C. Kappe, J. L. Redden, On the covering number of small alternating groups, Contemp. Math. 511 (2010) 109–125.
  • L. C. Kappe, D. Nikolova-Popova, E. Swartz, On the covering number of small symmetric groups and some sporadic simple groups, arXiv:1409.2292v1 [math.GR].
  • M. S. Lucido, On the covers of finite groups, in: C. M. Campbell, E. F. Robertson, G. C. Smith (Eds), Groups St. Andrews 2001, in Oxford, vol II, in : London Math. Soc. Lecture Note Ser. 305, 2003, 395–399.
  • A. Maróti, Covering the symmetric groups with proper subgroups, J. Combin. Theory Ser. A 110(1) (2005) 97–111.
  • M. J. Tomkinson, Groups as the union of proper subgroups, Math. Scand. 81(2) (1997) 191–198.
There are 11 citations in total.

Details

Journal Section Articles
Authors

Michael Epstein This is me

Spyros S. Magliveras This is me

Publication Date August 9, 2016
Published in Issue Year 2016 Volume: 3 Issue: 3

Cite

APA Epstein, M., & Magliveras, S. S. (2016). The covering number of $M_{24}$. Journal of Algebra Combinatorics Discrete Structures and Applications, 3(3), 155-158. https://doi.org/10.13069/jacodesmath.90728
AMA Epstein M, Magliveras SS. The covering number of $M_{24}$. Journal of Algebra Combinatorics Discrete Structures and Applications. August 2016;3(3):155-158. doi:10.13069/jacodesmath.90728
Chicago Epstein, Michael, and Spyros S. Magliveras. “The Covering Number of $M_{24}$”. Journal of Algebra Combinatorics Discrete Structures and Applications 3, no. 3 (August 2016): 155-58. https://doi.org/10.13069/jacodesmath.90728.
EndNote Epstein M, Magliveras SS (August 1, 2016) The covering number of $M_{24}$. Journal of Algebra Combinatorics Discrete Structures and Applications 3 3 155–158.
IEEE M. Epstein and S. S. Magliveras, “The covering number of $M_{24}$”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 3, no. 3, pp. 155–158, 2016, doi: 10.13069/jacodesmath.90728.
ISNAD Epstein, Michael - Magliveras, Spyros S. “The Covering Number of $M_{24}$”. Journal of Algebra Combinatorics Discrete Structures and Applications 3/3 (August 2016), 155-158. https://doi.org/10.13069/jacodesmath.90728.
JAMA Epstein M, Magliveras SS. The covering number of $M_{24}$. Journal of Algebra Combinatorics Discrete Structures and Applications. 2016;3:155–158.
MLA Epstein, Michael and Spyros S. Magliveras. “The Covering Number of $M_{24}$”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 3, no. 3, 2016, pp. 155-8, doi:10.13069/jacodesmath.90728.
Vancouver Epstein M, Magliveras SS. The covering number of $M_{24}$. Journal of Algebra Combinatorics Discrete Structures and Applications. 2016;3(3):155-8.