Cardinalities of irredundant bases of finite primitive groups

TL;DR

This paper demonstrates that for any interval of natural numbers, there exists a finite primitive group whose irredundant bases have exactly those cardinalities, revealing the diverse possible sizes of such bases.

Contribution

It establishes that the set of possible sizes of irredundant bases in finite primitive groups can be any interval of natural numbers, highlighting the variability in their structure.

Findings

Any interval of natural numbers can be realized as the set of irredundant base sizes.

The result applies specifically to finite primitive groups.

This expands understanding of base size variability in permutation groups.

Abstract

Let $G$ be a finite permutation group acting on a set $\Omega$. An ordered sequence $(\omega_1,\ldots,\omega_\ell)$ of elements of $\Omega$ is an irredundant base for $G$ if the pointwise stabilizer of the sequence is trivial and no point is fixed by the stabilizer of its predecessors. We show that any interval of natural numbers can be realized as the set of cardinalities of irredundant bases for some finite primitive group.