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# Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board Member: Blackburn, Simon R. / Conder, Marston / Dehornoy, Patrick / Eick, Bettina / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Schäge, Sven / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura

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# On the covering number of small symmetric groups and some sporadic simple groups

Luise-Charlotte Kappe
• Department of Mathematical Sciences, State University of New York at Binghamton, Binghamton, NY 13902-6000, United States of America
• :
/ Daniela Nikolova-Popova
• Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431, United States of America
• :
/ Eric Swartz
Published Online: 2016-10-12 | DOI: https://doi.org/10.1515/gcc-2016-0010

## Abstract

A set of proper subgroups is a cover for a group if its union is the whole group. The minimal number of subgroups needed to cover G is called the covering number of G, denoted by $\sigma \left(G\right)$. Determining $\sigma \left(G\right)$ is an open problem for many nonsolvable groups. For symmetric groups ${S}_{n}$, Maróti determined $\sigma \left({S}_{n}\right)$ for odd n with the exception of $n=9$ and gave estimates for n even. In this paper we determine $\sigma \left({S}_{n}\right)$ for $n=8,9,10,12$. In addition we find the covering number for the Mathieu group ${M}_{12}$ and improve an estimate given by Holmes for the Janko group ${J}_{1}$.

MSC 2010: 20D06; 20D60; 20D08; 20-04; 20F99

Published Online: 2016-10-12

Published in Print: 2016-11-01

Funding Source: Australian Research Council

Award identifier / Grant number: DP120101336

The third author acknowledges the support of the Australian Research Council Discovery Grant DP120101336 during his time spent at The University of Western Australia.

Citation Information: Groups Complexity Cryptology. Volume 8, Issue 2, Pages 135–154, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, October 2016