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

Document Type : Research Paper


1 University of Firenze

2 The University of Western Australia

3 University of Milano-Bicocca


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.


Main Subjects

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