Locally dihedral block designs and primitive groups with dihedral point stabilizers

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.