Locally dihedral block designs and primitive groups with dihedral point stabilizers
Jianfu Chen, Yanni Wu, Binzhou Xia
TL;DR
This work characterizes block designs with locally transitive automorphism groups whose local actions are dihedral, culminating in a complete classification of primitive groups with dihedral point stabilizers. The authors establish that almost simple and affine types exhaust the possibilities, with socles restricted to ${ m PSL}_2(q)$ or ${ m Sz}(q)$ in the almost simple case, and they derive explicit structures and intersection properties of relevant maximal subgroups. They then translate these group-theoretic classifications into structural results for symmetric and non-symmetric designs, proving faithfulness of local actions, conjugacy of point and block stabilizers in symmetric cases, and imprimitive behavior on points and blocks in many instances. As an application, they determine that flag-transitive symmetric designs with point stabilizers of order $2p$ (prime $p$) yield a unique $2$-$(16,6,2)$ design, highlighting the tight interaction between local group actions and global design parameters.
Abstract
Let $\mathcal{D}$ be a block design admitting a locally transitive automorphism group $G$. We say $\mathcal{D}$ is $G$-point-locally dihedral if the induced local action $G_x^{\mathcal{D}(x)}$ is dihedral for each point $x$, and say $\mathcal{D}$ is $G$-block-locally dihedral if the induced local action $G_B^B$ is dihedral for each block $B$. The design $\mathcal{D}$ is called $G$-locally dihedral if both conditions hold. We give a classification of primitive permutation groups with dihedral point stabilizers, and apply it to classify point-locally dihedral block designs. For symmetric designs with a dihedral local action, we show that $G_x$ and $G_B$ are conjugate in $G$. Moreover, both local actions are faithful, and $G$ acts imprimitively on both points and blocks.