Coprime Subdegrees of Twisted Wreath Permutation Groups

TL;DR

This paper constructs infinite families of primitive twisted wreath permutation groups with nontrivial coprime subdegrees, answering an open question and classifying such groups for specific parameters.

Contribution

It provides the first infinite families of primitive twisted wreath groups with nontrivial coprime subdegrees and classifies these subdegrees for certain cases.

Findings

Constructed infinite families of primitive twisted wreath groups with coprime subdegrees.

Determined all parameter values for which these groups have nontrivial coprime subdegrees.

Provided a full classification for the case m=2 and q not in {7,11,29}.

Abstract

Dolfi, Guralnick, Praeger and Spiga asked if there exist infinitely many primitive groups of twisted wreath type with nontrivial coprime subdegrees. Here we settle this question in the affirmative. We construct infinite families of primitive twisted wreath permutation groups with nontrivial coprime subdegrees. In particular, we define a primitive twisted wreath group $G(m,q)$ constructed from the nonabelian simple group $\text{PSL}(2,q)$ and a primitive permutation group of diagonal type with socle $\text{PSL}(2,q)^m$, and determine all values of $m$ and $q$ for which $G(m,q)$ has nontrivial coprime subdegrees. In the case where $m=2$ and $q\notin\{7,11,29\}$ we obtain a full classification of all pairs of nontrivial coprime subdegrees.