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 H1,,Hn, 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(Sn) for odd n9 and provided an estimate for even n. The second author later determined σ(Sn) for n 0(mod6) 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 Sn for various other small n. Currently, n = 14 is the smallest value for which σ(Sn) is unknown. In this paper, we prove the covering number of S14 is 3096.

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