Covers and Normal Covers of Finite Groups

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 $\gamma (G)\le (n+2)/2.$ 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 $\gamma (G)=2.$

Cited by in corpus (1)

• On Covers of Dihedral 2-Groups by Powerful Subgroups