Bases of twisted wreath products

Abstract

Finite quasiprimitive permutation groups of twisted wreath type are the finite permutation groups with a unique minimal normal subgroup which is non-abelian, non-simple and acts regularly. If T is a non-abelian simple group and P is a group that conveys transitive action on the set $\textbf{k}=\{1,2,\ldots ,k\}$ with $k\geqslant 2$, then every permutation group in this classification can be considered permutation isomorphic to $G=T^k{:}P$, a twisted wreath product acting on its base group $\Omega =T^k$. We prove that if $T\cong \textrm{A}_n,P\cong M^l{:}N\leqslant \textrm{S}_k$ with $M=\textrm{A}_s$ or classical group with dimensions less than or equal to $n-2$, $n\leqslant \{8,s,\ell \}$, then the base size of G is 2. Additionally, we demonstrate three possible values of the base size when P is semiprimitive on $\textbf{k}$ and G is quasiprimitive on $\Omega $.

Keywords

Quasiprimitive / Permutation group / Twisted wreath product / Base size / 20B05

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Funding

National Science Foundation(Grant No. 12301030)

Natural Science Foundation of China(12071376)

Natural Science Foundation Project of Chongqing, Chongqing Science and Technology Commission(CSTB2024NSCQ-MSX0544)

Doctoral Through Train Scientific Research Project of Chongqing(sl202100000324)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature

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