The endomorphism tower of a finite symmetric group

Abstract

We consider the endomorphism tower of a monoid $M$, that is, the sequence of monoids End${}_i(M)$ where End${}_0(M)=M$ and for all $i\ge 1$, End${}_i(M)$ is the monoid of all endomorphisms of End${}_{i-1}(M)$. We show that for a finite monoid $M$ this sequence does not stabilise in a finite number of steps. Our focus is then on the case where $M={\mathcal{S}}_n$, the symmetric group on a finite number $n$ of points. It is well known that other than in exceptional cases (which are avoided by taking $n\ge 7$), the corresponding automorphism tower of ${\mathcal{S}}_n$ stabilises at the first step. In spite of the natural nature of this question, nothing was known of the endomorphism tower above the level $i=1$. We determine (for each $n\ge 7)$ the elements of End${}_2({\mathcal{S}}_n)$ and their multiplication and thus verify that the monoids End${}_i({\mathcal{S}}_n)$ for $i=0,1,2$ all have group of units isomorphic to ${\mathcal{S}}_n$. We show that the same is true of End${}_3({\mathcal{S}}_n)$.