Irredundant bases for soluble groups

TL;DR

This paper establishes asymptotically tight bounds for the maximum length of irredundant bases in primitive soluble groups, disproves a related conjecture, and confirms Cameron's Greedy Conjecture for groups of odd order.

Contribution

It provides the first asymptotically tight bounds for irredundant bases in primitive soluble groups and resolves open conjectures in the field.

Findings

Derived asymptotically tight bounds for irredundant bases

Disproved Seress's conjecture on greedy bases

Proved Cameron's Greedy Conjecture for groups of odd order

Abstract

Let $\Delta$ be a finite set and $G$ be a subgroup of $\operatorname{Sym}(\Delta)$. An irredundant base for $G$ is a sequence of points of $\Delta$ yielding a strictly descending chain of pointwise stabilisers, terminating with the trivial group. Suppose that $G$ is primitive and soluble. We determine asymptotically tight bounds for the maximum length of an irredundant base for $G$. Moreover, we disprove a conjecture of Seress on the maximum length of an irredundant base constructed by the natural greedy algorithm, and prove Cameron's Greedy Conjecture for $|G|$ odd.