On the covering number of S14

Oppenheim, Ryan; Swartz, Eric

doi:10.2140/involve.2019.12.89

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On the covering number of $S_{14}$

Ryan Oppenheim and Eric Swartz

Vol. 12 (2019), No. 1, 89–96

DOI: 10.2140/involve.2019.12.89

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 $σ (G)$ . Maróti determined $σ (S_{n})$ for odd $n \neq 9$ and provided an estimate for even $n$ . The second author later determined $σ (S_{n})$ for $n \equiv 0 (mod 6)$ when $n \geq 18$ , 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 $σ (S_{n})$ is unknown. In this paper, we prove the covering number of $S_{14}$ is $3096$ .

Keywords

symmetric groups, finite union of proper subgroups, subgroup covering

Mathematical Subject Classification 2010

Primary: 20-04, 20D60

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Received: 9 July 2017

Revised: 28 November 2017

Accepted: 30 December 2017

Published: 31 May 2018

Communicated by Kenneth S. Berenhaut

Authors

Ryan Oppenheim
Department of Mathematics College of William and Mary Williamsburg, VA United States

Eric Swartz
Department of Mathematics College of William and Mary Williamsburg, VA United States

Vol. 12, No. 1, 2019