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

Abstract

Let $\gamma (S_n)$ be the minimum number of proper subgroups $H_i$ 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 $n$ 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 $({\alpha }_1,{\alpha }_2)\ne (1,1)$. 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.

17 pages

Cited by in corpus (2)

• Intersective $S_n$ polynomials with few irreducible factors
• Normal covering numbers for $S_n$ and $A_n$ and additive combinatorics