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- Title
Nilpotent covers of symmetric and alternating groups.
- Authors
Gill, Nick; Kimeu, Ngwava Arphaxad; Short, Ian
- 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.
- Subjects
MAXIMAL subgroups; PARTITIONS (Mathematics); CONJUGACY classes; INTEGERS
- Publication
Bulletin of the Belgian Mathematical Society - Simon Stevin, 2022, Vol 29, Issue 2, p249
- ISSN
1370-1444
- Publication type
Article
- DOI
10.36045/j.bbms.220218