Covers and Normal Covers of Finite Groups

TL;DR

This paper introduces a new measure for finite groups called $\gamma(G)$, explores its bounds for permutation groups, and investigates the structure of groups where this measure equals the minimal cover size or equals 2.

Contribution

It defines the invariant $\gamma(G)$ for finite groups, establishes bounds for permutation groups, and analyzes the structure of groups with specific $\gamma(G)$ values.

Findings

For noncyclic permutation groups of degree n, $\gamma(G)\leq (n+2)/2$.

Characterization of groups where $\gamma(G)=\sigma(G)$.

Identification of groups with $\gamma(G)=2$.

Abstract

For a finite non cyclic group $G$, let $\gamma(G)$ be the smallest integer $k$ such that $G$ contains $k$ proper subgroups $H_1,\dots,H_k$ with the property that every element of $G$ is contained in $H_i^g$ for some $i \in \{1,\dots,k\}$ and $g \in G.$ We prove that if $G$ is a noncyclic permutation group of degree $n,$ then $\gamma(G)\leq (n+2)/2.$ We then investigate the structure of the groups $G$ with $\gamma(G)=\sigma(G)$ (where $\sigma(G)$ is the size of a minimal cover of $G$) and of those with $\gamma(G)=2.$