Abstract
The normal covering number \(\gamma (G)\) of a finite, non-cyclic group G is the minimum number of proper subgroups such that each element of G lies in some conjugate of one of these subgroups. We find lower bounds linear in n for \(\gamma (S_n)\), when n is even, and for \(\gamma (A_n)\), when n is odd.
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References
Andrews, G.E.: The Theory of Partitions, Encyclopedia of Mathematics and its Applications, vol. 2. Addison-Wesley Publishing Co., Reading, MA (1976)
Bachraoui, M.E.: Relatively prime partitions with two and three parts. Fibonacci Q. 46/47, 341–345 (2008/09)
Brandl, R., Bubboloni, D., Hupp, I.: Polynomials with roots mod \(p\) for all primes \(p\). J. Group Theory 4, 233–239 (2001)
Britnell, J.R., Maróti, A.: Normal coverings of linear groups. Algebra Number Theory 7, 2085–2102 (2013)
Bubboloni, D., Praeger, C.E.: Normal coverings of the alternating and symmetric groups. J. Comb. Theory Ser. A 118, 2000–2024 (2011)
Bubboloni, D., Praeger, C.E., Spiga, P.: Normal coverings and pairwise generation of finite alternating and symmetric groups. J. Algebra 390, 199–215 (2013)
Bubboloni, D., Praeger, C.E., Spiga, P.: Conjectures on the normal covering number of finite alternating and symmetric groups. IJGT 3(2), 57–75 (2014)
Bubboloni, D., Luca, F., Spiga, P.: Compositions of \(n\) satisfying some coprimality conditions. J. Number Theory 132, 2922–2946 (2012)
Bubboloni, D., Sonn, J.: Intersective \(S_n\) polynomials with few irreducible factors. Manuscr. Math. 151, 477–492 (2016)
Bugeaud, Y., Shorey, T.N.: On the Diophantine equation \(\frac{x^m-1}{x-1}=\frac{y^n-1}{y-1}\). Pac. J. Math. 207, 61–75 (2002)
Guest, S., Morris, J., Praeger, C.E., Spiga, P.: Affine transformations of finite vector spaces with large orders or few cycles. J. Pure Appl. Algebra 219, 308–330 (2015)
Guest, S., Morris, J., Praeger, C.E., Spiga, P.: Finite primitive permutation groups containing a permutation having at most four cycles. J. Algebra 454, 233–251 (2016)
Khukhro, E.I., Mazurov, V.D.: Unsolved problems in Group Theory. The Kourovka Notebook. arXiv:1401.0300v13 [math.GR]
Maróti, A.: Private communication by an email to the authors
Niven, Ivan, Zuckerman, Herbert S., Montgomery, Hugh L.: An Introduction to the Theory of Numbers, 5th edn. Wiley, New York (1991)
Rabayev, D., Sonn, J.: On Galois realizations of the \(2\)-coverable symmetric and alternating groups. Commun. Algebra 42, 253–258 (2014)
Acknowledgements
We are profoundly in debted to Attila Maróti for his insights into this problem and for permitting us to use his personal communication.
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Communicated by J. S. Wilson.
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The first author is partially supported by GNSAGA of INdAM. The second author is supported by Australian Research Council Discovery Project Grant DP130100106.
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Bubboloni, D., Praeger, C.E. & Spiga, P. Linear bounds for the normal covering number of the symmetric and alternating groups. Monatsh Math 191, 229–247 (2020). https://doi.org/10.1007/s00605-019-01287-5
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DOI: https://doi.org/10.1007/s00605-019-01287-5