Bases of twisted wreath products
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 nonabelian, nonsimple 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 nonabelian 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2021
 DOI:
 10.48550/arXiv.2102.02190
 arXiv:
 arXiv:2102.02190
 Bibcode:
 2021arXiv210202190F
 Keywords:

 Mathematics  Group Theory;
 20B15;
 20B05
 EPrint:
 A false statement has been removed from Lemma 4.9 and Theorem 8.1, and a more general version of Theorem 8.1 is given (with the same proof). Minor revisions have been made throughout. 18 pages