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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