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Normal coverings of finite symmetric and alternating groups
Authors:Daniela Bubboloni
Affiliation:a Dipartimento di Matematica per le Decisioni, Università di Firenze, via Lombroso 6/17, 50134 Firenze, Italy
b Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia
Abstract:In this paper we investigate the minimum number of maximal subgroups Hi, i=1,…,k of the symmetric group Sn (or the alternating group An) such that each element in the group Sn (respectively An) lies in some conjugate of one of the Hi. We prove that this number lies between a?(n) and bn for certain constants a,b, where ?(n) is the Euler phi-function, and we show that the number depends on the arithmetical complexity of n. Moreover in the case where n is divisible by at most two primes, we obtain an upper bound of 2+?(n)/2, and we determine the exact value for Sn when n is odd and for An when n is even.
Keywords:Covering   Symmetric group   Alternating group
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