Bases of twisted wreath products

TL;DR

This paper investigates the base sizes of finite quasiprimitive permutation groups of twisted wreath type, establishing conditions for base size 2, asymptotic behavior of base pairs, and classifying possible base sizes.

Contribution

It provides new results on base sizes of twisted wreath type groups, including exact values, asymptotic properties, and classifications up to four possible base sizes.

Findings

Primitive twisted wreath groups have base size 2.

Proportion of bases tends to 1 as group size increases.

Many primitive groups of this type have arbitrarily large base sizes.

Abstract

We study the base sizes of finite quasiprimitive permutation groups of twisted wreath type, which are precisely the finite permutation groups with a unique minimal normal subgroup that is also non-abelian, non-simple and regular. Every permutation group of twisted wreath type is permutation isomorphic to a twisted wreath product $G=T^k{:}P$ acting on its base group $\Omega=T^k$, where $T$ is some non-abelian simple group and $P$ is some group acting transitively on $\boldsymbol{k}=\{1,\ldots,k\}$ with $k\geq 2$. We prove that if $G$ is primitive on $\Omega$ and $P$ is quasiprimitive on $\boldsymbol{k}$, then $G$ has base size 2. We also prove that the proportion of pairs of points that are bases for $G$ tends to 1 as $|G|\to \infty$ when $G$ is primitive on $\Omega$ and $P$ is primitive on $\boldsymbol{k}$. Lastly, we determine the base size of any quasiprimitive group of twisted wreath type up to four possible values (and three in the primitive case). In particular, we demonstrate that there are many families of primitive groups of twisted wreath type with arbitrarily large base sizes.