  The normal covering number $\gamma(G)$ of a finite, non-cyclic group $G$ is
the minimum number of proper subgroups such that each element of $G$ lies in
some conjugate of one of these subgroups. We find lower bounds linear in $n$
for $\gamma(S_n)$, when $n$ is even, and for $\gamma(A_n)$, when $n$ is odd.
