Author/Authors :
بوبولوني، دانيلا نويسنده Dipartimento di Scienze per lEconomia e lImpresa, University of Firenze, Via delle Pandette 9-D6, 50127 Firenze, Italy Bubboloni, Daniela , پريگر، شريل اي. نويسنده Centre for Mathematics of Symmetry and Computation, School of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Austr Praeger, Cheryl E. , اسپيگا، پابلو نويسنده Dipartimento di Matematica e Applicazioni, University of Milano-Bicocca, Via Cozzi 53, 20125 Milano, Italy Spiga, Pablo
Abstract :
Let gamma(Sn) be the minimum number of proper subgroups Hi, i =
1,...,ell, of the symmetric group Sn such that each element in Sn lies in some
conjugate of one of the Hi. In this paper we conjecture that
gamma(Sn) =(n/2)(1-1/p_1) (1-1/p_2) + 2,
where p1, p2 are the two smallest primes in the factorization of n and n is
neither a prime power nor a product of two primes. Support for the conjecture
is given by a previous result for the case where n has at most two distinct
prime divisors. We give further evidence by confirming the conjecture for
certain integers of the form n = 15q, for an infinite set of primes q, and by
reporting on a Magma computation. We make a similar conjecture for
gamma(An), when n is even, and provide a similar amount of evidence.
