فهرست نشریات علمی دسترسی آزاد ایران

Let $\gamma(S_n)$ be the minimum number of proper subgroups‎ ‎$H_i,\ i=1‎, ‎\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 $$\gamma(S_n)=\frac{n}{2}\left(1-\frac{1}{p_1}\right)‎ ‎\left(1-\frac{1}{p_2}\right)+2,$$ where $p_1,p_2$ 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 $n=p_1^{\alpha_1}p_2^{\alpha_2},$ with $(\alpha_1,\alpha_2)\neq (1,1)$‎. ‎We give further evidence by confirming the conjecture for 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 $\gamma(A_n)$‎, ‎when $n$ is even‎, ‎and provide a similar amount of evidence‎.