Locally dihedral block designs and primitive groups with dihedral point stabilizers
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915935733940224 |
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| author | Chen, Jianfu Wu, Yanni Xia, Binzhou |
| author_facet | Chen, Jianfu Wu, Yanni Xia, Binzhou |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_08678 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Locally dihedral block designs and primitive groups with dihedral point stabilizers Chen, Jianfu Wu, Yanni Xia, Binzhou Combinatorics Group Theory 20B15, 20B25, 20E28, 05E18 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. |
| title | Locally dihedral block designs and primitive groups with dihedral point stabilizers |
| topic | Combinatorics Group Theory 20B15, 20B25, 20E28, 05E18 |
| url | https://arxiv.org/abs/2601.08678 |