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Normal covering numbers for $S_n$ and $A_n$ and additive combinatorics
The normal covering number $\gamma(G)$ of a finite group $G$ is the minimum number of proper subgroups whose conjugates cover the group. We give various estimates for $\gamma(S_n)$ and $\gamma(A_n)$ depending on the arithmetic structure of $n$. In particular we determine the limsups over $\gamma(S_n) / n$ and $\gamma(A_n) / …