Block-Transitive $5−(v,k,2)$ Designs with k divides v
Abstract
The classification of block-transitive 5-designs remains an open problem. The additional parameter condition that k divides v is called the Camina-Gagen condition. In this paper, we investigate block-transitive simple $5−(v,k,2)$ designs satisfying the Camina-Gagen condition. Using the classification of finite $2$-homogeneous permutation groups, we consider the affine and almost simple cases separately. We prove that no such design admits a block-transitive automorphism group of affine type. For the almost simple case, up to isomorphism, there are exactly two possibilities: a $5−(12,6,2)$ design admitting PGL(2,11) as a block-transitive automorphism group and a $5−(24,8,2)$ design admitting PGL(2,23) as a block-transitive automorphism group.
Related
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BibTeX
@misc{hu2026blocktransitive,
title = {{Block-Transitive $5$-$(v,k,2)$ Designs with $k$ divides $v$}},
author = {Huijiao Hu and Zheng Huang and Shouqiang Shen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36613},
eprint = {2609.36613},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36613},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Block-Transitive $5−(v,k,2)$ Designs with k divides v
AU - Huijiao Hu
AU - Zheng Huang
AU - Shouqiang Shen
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36613
UR - https://arxiv.org/abs/2609.36613
AB - The classification of block-transitive 5-designs remains an open problem. The additional parameter condition that k divides v is called the Camina-Gagen condition. In this paper, we investigate block-transitive simple $5−(v,k,2)$ designs satisfying the Camina-Gagen condition. Using the classification of finite $2$-homogeneous permutation groups, we consider the affine and almost simple cases separately. We prove that no such design admits a block-transitive automorphism group of affine type. For the almost simple case, up to isomorphism, there are exactly two possibilities: a $5−(12,6,2)$ design admitting PGL(2,11) as a block-transitive automorphism group and a $5−(24,8,2)$ design admitting PGL(2,23) as a block-transitive automorphism group.
ER -