arXiv:2005.13869v1 [math.GR] 28 May 2020 NILPOTENT COVERS OF SYMMETRIC GROUPS KIMEU ARPHAXAD NGWAVA, NICK GILL, AND IAN SHORT 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 order of M. In addition, we prove that the order of M is equal to the order 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. These results contrast starkly with those for abelian covers of Sn. 1. Introduction The principal objective of this paper is to determine the least number of nilpotent subgroups of the symmetric group on n letters S n that are necessary to cover S n . We establish that there is a unique minimal collection of maximal nilpotent subgroups that cover S n , and the order of this collection can be computed easily from a list of the partitions of n into distinct positive integers. Furthermore, we prove that the order of the collection is equal to the order of a maximal non-nilpotent subset of S n . These results are stronger than similar results for abelian covers, suggesting that nilpotent covers are a richer class of covers to study. To explain our results in more detail, consider a finite group G.A nilpotent cover of G is a finite family M of nilpotent subgroups of G for which G =  H∈M H. A nilpotent cover M of G is said to be minimal if no other nilpotent cover of G has fewer members. Let Σ ∞ (G) denote the size of a minimal nilpotent cover of G, provided such a cover exists. Of particular interest to us are nilpotent covers that are invariant under conjugation. A nilpotent cover M of G is normal if whenever H ∈M and g ∈ G we have g −1 Hg ∈M. We seek to ascertain whether a minimal nilpotent cover of G can be found that is normal. Each normal nilpotent cover of G can be partitioned into conjugacy classes of subgroups. Let Γ ∞ (G) denote the least number of such conjugacy classes, among all the normal nilpotent covers of G. There is a parallel notion to that of nilpotent cover: a non-nilpotent subset of G is a subset X of G such that for any two distinct elements x and y of X , the subgroup 〈x, y〉 generated by x and y is not nilpotent. A non-nilpotent subset of G is said to be maximal if no other non-nilpotent subset of G contains more elements. Let σ ∞ (G) denote the size of a maximal non-nilpotent subset of G. A straightforward consequence of the pigeon-hole principle is that σ ∞ (G) ≤ Σ ∞ (G), provided G has a nilpotent cover. In this paper we calculate the quantities Γ ∞ (S n ), Σ ∞ (S n ) and σ ∞ (S n ), and prove that the latter two quantities coincide. We use the concept of a distinct partition of a positive integer n, which is a set T = {t 1 ,t 2 ,...,t k } where t 1 ,t 2 ,...,t k are distinct positive integers and n = t 1 + t 2 + ··· + t k . In Section 2 we will prove that if the cycle type of an element g of S n is a distinct partition of n, then g lies within a unique maximal nilpotent subgroup of S n (Proposition 2.4). We denote by M the collection of all maximal nilpotent subgroups that arise in this way. Theorem 1.1. The cover M is the unique minimal nilpotent cover of S n by maximal nilpotent subgroups. The cover M is by definition a normal cover, so we obtain the following corollary of Proposition 2.4 and Theorem 1.1. Corollary 1.2. The cover M is a normal nilpotent cover of S n and Γ ∞ (S n ) is equal to the number of partitions of n into distinct positive integers. Key words and phrases. alternating group; nilpotent cover; non-nilpotent subset; normal nilpotent cover; symmetric group. 1