Covering the set of p-elements in finite groups by proper subgroups

Let p be a prime and let G be a finite group which is generated by the set $G_p$ of its p-elements. We show that if G is solvable and not a p-group, then the minimal number ${\sigma }_p\left(G\right)$ of proper subgroups of G whose union contains $G_p$ is equal to 1 less than the minimal number of proper subgroups of G whose union is G. For p-solvable groups G, we always have ${\sigma }_p\left(G\right)\ge p+1$. We study the case of alternating and symmetric groups G in detail.