Irredundant bases for soluble groups

Let Δ$\Delta$ be a finite set and G$G$ be a subgroup of Sym(Δ)$\operatorname{Sym}(\Delta)$. An irredundant base for G$G$ is a sequence of points of Δ$\Delta$ yielding a strictly descending chain of pointwise stabilisers, terminating with the trivial group. Suppose that G$G$ is primitive and soluble....

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Published inThe Bulletin of the London Mathematical Society Vol. 57; no. 10; pp. 3013 - 3023
Main Authors Brenner, Sofia, del Valle, Coen, Roney‐Dougal, Colva M.
Format Journal Article
LanguageEnglish
Published 01.10.2025
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ISSN0024-6093
1469-2120
1469-2120
DOI10.1112/blms.70137

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Summary:Let Δ$\Delta$ be a finite set and G$G$ be a subgroup of Sym(Δ)$\operatorname{Sym}(\Delta)$. An irredundant base for G$G$ is a sequence of points of Δ$\Delta$ yielding a strictly descending chain of pointwise stabilisers, terminating with the trivial group. Suppose that G$G$ is primitive and soluble. We determine asymptotically tight bounds for the maximum length of an irredundant base for G$G$. 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|$|G|$ odd.
ISSN:0024-6093
1469-2120
1469-2120
DOI:10.1112/blms.70137