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, 29(2) pp. 249–266.

DOI: https://doi.org/10.36045/j.bbms.220228

URL: https://projecteuclid.org/journals/bulletin-of-the...

Abstract

We prove that the symmetric group S_n 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 S_n. 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 A_n. In particular, we prove that, for all but finitely many values of _n, no minimal cover of A_n by maximal nilpotent subgroups is a normal cover and the size of a minimal cover of A_n by maximal nilpotent subgroups is strictly greater than the size of a maximal non-nilpotent subset of A_n.

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• Item ORO ID
• 85242
• Item Type
• Journal Item
• Project Funding Details

• Funded Project Name	Project ID	Funding Body
  The Open University (OU)	Not Set	The Open University (OU)
  Mentoring African Researchers in Mathematics	Not Set	LMS, 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.
  
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  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