On the Intersection Numbers of Finite Groups

TL;DR

This paper introduces the intersection number of finite groups, exploring its properties, calculating it for nilpotent groups, and analyzing specific non-nilpotent families, thereby expanding understanding of subgroup intersections.

Contribution

It defines the intersection number for finite groups, provides an exact formula for nilpotent groups, and investigates this invariant in certain non-nilpotent group families.

Findings

Exact formula for intersection number in nilpotent groups

Determined intersection numbers for specific non-nilpotent groups

Discussed generalizations and open questions about the invariant

Abstract

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.