Irredundant bases for the symmetric group

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 $S_n$ or $A_n$ 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.

References in corpus (2)

• On relational complexity and base size of finite primitive groups
• Irredundant bases for finite groups of Lie type