On the covering number of $S_{14}$

Abstract

If all elements of a group $G$ are contained in the set-theoretic union of proper subgroups $H_1,\dots ,H_n$, then we define this collection to be a cover of $G$. When such a cover exists, the cardinality of the smallest possible cover is called the covering number of $G$, denoted by $\sigma \left(G\right)$. Maróti determined $\sigma \left(S_n\right)$ for odd $n\ne 9$ and provided an estimate for even $n$. The second author later determined $\sigma \left(S_n\right)$ for $n\equiv 0\left(mod6\right)$ when $n⩾18$, while joint work of the second author with Kappe and Nikolova-Popova also verified that Maróti’s rule holds for $n=9$ and established the covering numbers of $S_n$ for various other small $n$. Currently, $n=14$ is the smallest value for which $\sigma \left(S_n\right)$ is unknown. In this paper, we prove the covering number of $S_{14}$ is $3096$.