Conjectures on the normal covering number of finite symmetric and alternating groups

Let \gamma(S_n) be the minimum number of proper subgroups‎ ‎H_i,\ i=۱‎, ‎\dots‎, ‎l of the symmetric group S_n such that each element in S_n‎ ‎lies in some conjugate of one of the H_i. In this paper we‎ ‎conjecture that \gamma(S_n)=\frac{n}{۲}\left(۱-\frac{۱}{p_۱}\right)‎ ‎\left(۱-\frac{۱}{p_۲}\right)+۲, where p_۱,p_۲ are the two smallest primes‎ ‎in the factorization of n\in\mathbb{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 n=p_۱^{\alpha_۱}p_۲^{\alpha_۲}, with (\alpha_۱,\alpha_۲)\neq (۱,۱)‎. ‎We give further evidence by confirming the conjecture for integers‎ ‎of the form n=۱۵q for an infinite set of primes q‎, ‎and by reporting on a‎ ‎ Magma computation‎. ‎We make a similar conjecture for \gamma(A_n)‎, ‎when n is even‎, ‎and provide a similar amount of evidence‎.

کلیدواژه ها:

Covering ، symmetric group ، alternating group