Normal $2$-coverings of the finite simple groups and their generalizations

Abstract: Given a finite group $G$, we say that $G$ has weak normal covering number $\gamma_w(G)$ if $\gamma_w(G)$ is the smallest integer with $G$ admitting proper subgroups $H_1,\ldots,H_{\gamma_w(G)}$ such that each element of $G$ has a conjugate in $H_i$, for some $i\in {1,\ldots,\gamma_w(G)}$, via an element in the automorphism group of $G$. We prove that the weak normal covering number of every non-abelian simple group is at least $2$ and we classify the non-abelian simple groups attaining $2$. As an application, we classify the non-abelian simple groups having normal covering number $2$. We also show that the weak normal covering number of an almost simple group is at least two up to one exception. We determine the weak normal covering number and the normal covering number of the almost simple groups having socle a sporadic simple group. Using similar methods we find the clique number of the invariably generating graph of the almost simple groups having socle a sporadic simple group.