Large minimal invariable generating sets in the finite symmetric groups

TL;DR

This paper investigates the maximum size of minimal invariable generating sets in finite symmetric groups, establishing bounds that show it grows asymptotically like half the degree of the group.

Contribution

It provides the first bounds on the size of minimal invariable generating sets in symmetric groups, demonstrating their growth rate as the group order increases.

Findings

Bounds for $m_I(S_n)$ established

Asymptotic growth of $m_I(S_n)$ shown to be $n/2$

Results contribute to understanding invariable generation in symmetric groups

Abstract

For a finite group $G$, let $m_I(G)$ denote the largest possible cardinality of a minimal invariable generating set of $G$. We prove an upper and a lower bound for $m_I(S_n)$, which show in particular that $m_I(S_n)$ is asymptotic to $n/2$ as $n\rightarrow \infty$.