#### Vol. 12, No. 1, 2019

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On the covering number of $S_{14}$

### Ryan Oppenheim and Eric Swartz

Vol. 12 (2019), No. 1, 89–96
##### 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 $n\equiv 0\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}6\right)$ when $n\ge 18$, 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 $3096$.

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##### Keywords
symmetric groups, finite union of proper subgroups, subgroup covering
##### Mathematical Subject Classification 2010
Primary: 20-04, 20D60

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