Invariable generation of permutation and linear groups

TL;DR

This paper investigates the minimal number of elements needed to invariably generate finite linear and permutation groups, providing new bounds and tighter results for specific cases.

Contribution

It establishes new upper bounds on the number of elements required to invariably generate finite linear and permutation groups, including specific bounds for fields of size 2 or 3.

Findings

Finite linear groups of dimension n can be invariably generated by approximately 1.5n elements.

Transitive permutation groups of degree n can be invariably generated by O(n / sqrt(log n)) elements.

Primitive permutation groups of degree n can be invariably generated by O(log n / sqrt(log log n)) elements.

Abstract

A subset $\left\{x_{1},x_{2},\hdots,x_{d}\right\}$ of a group $G$ \emph{invariably generates} $G$ if $\left\{x_{1}^{g_{1}},x_{2}^{g_{2}},\hdots,x_{d}^{g_{d}}\right\}$ generates $G$ for every $d$-tuple $(g_{1},g_{2}\hdots,g_{d})\in G^{d}$. We prove that a finite completely reducible linear group of dimension $n$ can be invariably generated by $\left\lfloor \frac{3n}{2}\right\rfloor$ elements. We also prove tighter bounds when the field in question has order $2$ or $3$. Finally, we prove that a transitive [respectively primitive] permutation group of degree $n\geq 2$ [resp. $n\geq 3$] can be invariably generated by $O\left(\frac{n}{\sqrt{\log{n}}}\right)$ [resp. $O\left(\frac{\log{n}}{\sqrt{\log{\log{n}}}}\right)$] elements.