On the maximal number of elements pairwise generating the symmetric group of even degree

TL;DR

This paper investigates the maximal size of subsets of the symmetric group of even degree where each pair generates the entire group, establishing asymptotic equivalences and bounds independent of the classification of finite simple groups.

Contribution

It provides asymptotic formulas for the maximal pairwise generating subsets and the minimal covering families in symmetric groups of even degree, including new bounds independent of simple group classification.

Findings

oth unctions re asymptotically qual to rac{1}{2} inom{n}{n/2} for even n.

 lower bound of (1-o(1))n on oth unctions, independent of simple group classification.

alculation of the clique number for a graph defined on the symmetric group for large n.

Abstract

Let $G$ be the symmetric group of degree $n$. Let $\omega(G)$ be the maximal size of a subset $S$ of $G$ such that $\langle x,y \rangle = G$ whenever $x,y \in S$ and $x \neq y$ and let $\sigma(G)$ be the minimal size of a family of proper subgroups of $G$ whose union is $G$. We prove that both functions $\sigma(G)$ and $\omega(G)$ are asymptotically equal to $\frac{1}{2} \binom{n}{n/2}$ when $n$ is even. This, together with a result of S. Blackburn, implies that $\sigma(G)/\omega(G)$ tends to $1$ as $n \to \infty$. Moreover, we give a lower bound of $(1-o(1))n$ on $\omega(G)$ which is independent of the classification of finite simple groups. We also calculate, for large enough $n$, the clique number of the graph defined as follows: the vertices are the elements of $G$ and two vertices $x,y$ are connected by an edge if $\langle x,y \rangle \geq A_n$.