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

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 $⟨x,y⟩=G$ whenever $x,y\in S$ and $x\ne y$ 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 $\frac{1}{2}\left(\genfrac{}{}{0ex}{}{n}{n/2}\right)$ 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 $⟨x,y⟩\ge A_n$.