A \emph{ finite cover} of a group $G$ is a finite collection $\mathcal{ C}$ of proper subgroups of $G$ with the property that $\bigcup \mathcal{ C} = G$. A finite group admits a finite cover if and only if it is noncyclic. More generally, it is known that a group admits a finite cover if and only if it has a finite, noncyclic homomorphic image. If $\mathcal{ C}$ is a finite cover of a group $G$, and no cover of $G$ with fewer subgroups exists, then $\mathcal{ C}$ is said to be a \emph{ minimal cover} of $G$, and the cardinality of $\mathcal{ C}$ is called the \emph{ covering number} of $G$, denoted by $\sigma(G)$. Here we investigate the covering numbers of the McLaughlin sporadic simple group and some low degree primitive groups.
Categories: math.GR math.CO