On the maximal number of elements pairwise generating the finite alternating group

TL;DR

This paper investigates the maximal size of subsets of the alternating group where pairwise elements generate the entire group, and compares it to the minimal covering by proper subgroups, revealing asymptotic behavior and explicit calculations.

Contribution

It establishes that the ratio of the minimal subgroup cover size to the maximal pairwise generating subset size approaches one for composite degrees, and provides explicit values for certain cases.

Findings

The ratio $\sigma(G)/\omega(G)$ tends to 1 as $n$ increases among composite numbers.

Explicit calculation of $\sigma(A_n)$ for $n \\equiv 3 \\mod 18$ and $n \\geq 21$.

Asymptotic equivalence of subgroup cover size and pairwise generating subset size.

Abstract

Let $G$ be the alternating group of degree $n$. Let $\omega(G)$ be the maximal size of a subset $S$ of $G$ such that $\langle x,y \rangle = G$ whenever $x,y \in S$ and $x \neq y$ and let $\sigma(G)$ be the minimal size of a family of proper subgroups of $G$ whose union is $G$. We prove that, when $n$ varies in the family of composite numbers, $\sigma(G)/\omega(G)$ tends to $1$ as $n \to \infty$. Moreover, we explicitly calculate $\sigma(A_n)$ for $n \geq 21$ congruent to $3$ modulo $18$.