Nilpotent covers of symmetric and alternating groups

Gill, Nick; Kimeu, Ngwava and Short, Ian (2022). Nilpotent covers of symmetric and alternating groups. Bulletin of the Belgian Mathematical Society – Simon Stevin (in press).

Abstract

We prove that the symmetric group Sn has a unique minimal cover M by maximal nilpotent subgroups, and we obtain an explicit and easily computed formula for the size of M. In addition, we prove that the size of M is equal to the size of a maximal non-nilpotent subset of Sn. This cover M has attractive properties; for instance, it is a normal cover, and the number of conjugacy classes of subgroups in the cover is equal to the number of partitions of n into distinct positive integers.

We show that these results contrast with those for the alternating group An. In particular, we prove that, for all but finitely many values of n, no minimal cover of An by maximal nilpotent subgroups is a normal cover and the size of a minimal cover of An by maximal nilpotent subgroups is strictly greater than the size of a maximal non-nilpotent subset of An.

Viewing alternatives

Item Actions

Export

About

  • Item ORO ID
  • 85242
  • Item Type
  • Journal Item
  • Project Funding Details
  • Funded Project NameProject IDFunding Body
    The Open University (OU)Not SetThe Open University (OU)
    Mentoring African Researchers in MathematicsNot SetLMS, IMU, AMMSI
  • Extra Information
  • Acceptance message on 21/03/2022:

    Dear Nick,

    I am pleased to inform you that your article

    Nilpotent covers of symmetric and alternating groups
    by Nick Gill, Kimeu Ngwava and Ian Short

    will be accepted by the editorial board for publication in the Bulletin of the Belgian Mathematical Society – Simon Stevin.

    Please add the AMS classification numbers (2020 version, see https://mathscinet-ams-org.myezproxy.vub.ac.be/msnhtml/msc2020.pdf). Then go to

    https://ef.msp.org/uploadsource.php?p_id=133018&cr=5FB19013EE

    and upload the texfile of your paper. Also, please print, complete and sign the enclosed Author's Agreement Form and send it to bull.bms@vub.be. More info about this form is appended below.

    You will receive digital proofs as the publication date nears, and at that time you will have a chance to correct typos and update the bibliography.

    Your paper will be officially accepted as soon as we get the .tex file and the Author's Agreement Form. On request we send an official letter of acceptance.

    Finally, please keep EditFlow informed at editflow@msp.org of any changes in your email address.

    Please do not reply to this message. You can track the status of your article and contact the journal or editors at

    https://ef.msp.org/status.php?p_id=133018&cr=5FB19013EE


    Sincerely,

    Stefaan Caenepeel
    Editor-in-Chief
  • Keywords
  • alternating group; nilpotent cover; non-nilpotent subset; normal nilpotent cover; symmetric group
  • Academic Unit or School
  • Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
    Faculty of Science, Technology, Engineering and Mathematics (STEM)
  • Copyright Holders
  • © 2022 The Belgian Mathematical Society
  • Depositing User
  • Ian Short

Recommendations