On the Intersection Numbers of Finite Groups

The covering number of a nontrivial finite group $G$, denoted $\sigma(G)$, is the smallest number of proper subgroups of $G$ whose set-theoretic union equals $G$. In this article, we focus on a dual problem to that of covering numbers of groups, which involves maximal subgroups of finite groups. For a nontrivial finite group $G$, we define the intersection number of $G$, denoted $\iota(G)$, to be the minimum number of maximal subgroups whose intersection equals the Frattini subgroup of $G$. We elucidate some basic properties of this invariant, and give an exact formula for $\iota(G)$ when $G$ is a nontrivial finite nilpotent group. In addition, we determine the intersection numbers of a few infinite families of non-nilpotent groups. We conclude by discussing a generalization of the intersection number of a nontrivial finite group and pose some open questions about these invariants.