On the Primary Coverings of Finite Solvable and Symmetric Groups

Abstract

A primary covering of a finite group $G$ is a family of proper subgroups of $G$ whose union contains the set of elements of $G$ having order a prime power. We denote with ${\sigma }_0(G)$ the smallest size of a primary covering of $G$, and call it the primary covering number of $G$. We study this number and compare it with its analogous $\sigma (G)$, the covering number, for the classes of groups $G$ that are solvable and symmetric.