The Covering Numbers of \(\mathbb{A}_{9}\) and \(\mathbb{A}_{11}\)

A collection \(\mathcal{S}\) of proper subgroups of a group \(G\) is said to be a cover (or covering) for \(G\) if the union of the members of \(\mathcal{S}\) is all of \(G\). A cover \(\mathcal{C}\) of minimal cardinality is called a minimal cover for \(G\) and \(|\mathcal{C}|\) is called the covering number of \(G\), denoted by \(\sigma(G)\). In this paper we determine the covering numbers of the alternating groups \(A_9\) and \(A_{11}\).