
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 acting on its base group , where is some non-abelian simple group and is some group acting transitively on with . We prove that if is primitive on and is quasiprimitive on , then has base size 2. We also prove that the proportion of pairs of points that are bases for tends to 1 as when is primitive on and is primitive on . Lastly, we determine the base size of any quasiprimitive group of twisted wreath…
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