Covers and Normal Covers of Finite Groups

For a finite non cyclic group $G$, let $γ(G)$ 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 $γ(G)\leq (n+2)/2.$ We then investigate the structure of the groups $G$ with $γ(G)=σ(G)$ (where $σ(G)$ is the size of a minimal cover of $G$) and of those with $γ(G)=2.$

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