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 $\langle x,y \rangle = G$ 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 $\langle x,y \rangle \geq A_n$.