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 $\mathcal{D}$ 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.