Locally dihedral block designs and primitive groups with dihedral point stabilizers

TL;DR

This paper classifies certain symmetric block designs with dihedral local symmetry, revealing that their automorphism groups are either imprimitive or Frobenius groups of odd order.

Contribution

It provides a classification of primitive groups with dihedral point stabilizers and applies this to characterize point-locally dihedral block designs.

Findings

In symmetric designs with dihedral or abelian local action, point and block stabilizers are conjugate.

Such groups are either imprimitive or Frobenius groups of odd order.

Abstract

Let $\mathcal{D}$ be a block design admitting a locally transitive automorphism group $G$. We say that $\mathcal{D}$ is $G$-point-locally dihedral if the induced local action $G_x^{\mathcal{D}}$ is dihedral for each point $x$, and that $\mathcal{D}$ is $G$-block-locally dihedral if the induced local action $G_B^B$ is dihedral for each block $B$. If both conditions hold, $\mathcal{D}$ is called $G$-locally dihedral. We give a classification of primitive permutation groups with dihedral point stabilizers and apply this to classify point-locally dihedral block designs. In particular, for symmetric designs with a dihedral or abelian local action, we show that $G_x$ and $G_B$ are conjugate in $G$, and that either $G$ acts imprimitively on both points and blocks, or $G$ is a Frobenius group of odd order.