Irredundant bases for the symmetric group

TL;DR

This paper establishes tight bounds on the maximum size of irredundant bases for symmetric and alternating groups acting primitively, showing they grow at most as the square root or square of the logarithm of n.

Contribution

It proves new upper bounds on the size of irredundant bases for primitive actions of symmetric and alternating groups, demonstrating these bounds are optimal.

Findings

Maximum irredundant base size is O(√n) for symmetric and alternating groups.

In most cases, the bound reduces to O((log n)^2).

These bounds are proven to be tight and optimal.

Abstract

An irredundant base of a group $G$ acting faithfully on a finite set $\Gamma$ is a sequence of points in $\Gamma$ that produces a strictly descending chain of pointwise stabiliser subgroups in $G$, terminating at the trivial subgroup. Suppose that $G$ is $\operatorname{S}_n$ or $\operatorname{A}_n$ acting primitively on $\Gamma$, and that the point stabiliser is primitive in its natural action on $n$ points. We prove that the maximum size of an irredundant base of $G$ is $O\left(\sqrt{n}\right)$, and in most cases $O\left((\log n)^2\right)$. We also show that these bounds are best possible.