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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2504.11407 |
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| _version_ | 1866912329937977344 |
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| author | Montinaro, Alessandro |
| author_facet | Montinaro, Alessandro |
| contents | A famous result of Higman and McLaughlin \cite{HM} in 1961 asserts that any flag-transitive automorphism group $G$ of a $2$-design $\mathcal{D}$ with $λ=1$ acts point-primitively on $\mathcal{D}$. In this paper, we show that the Higman and McLaughlin theorem is still true when $λ$ is a prime and $\mathcal{D}$ is not isomorphic to one of the two $2$-$(16,6,2)$ designs as in [42, Section 1.2], or the $2$-$(45,12,3)$ design as in [44, Construction 4.2], or, when $2^{2^{j}}+1$ is a Fermat prime, a possible $2$-$(2^{2^{j+1}}(2^{2^{j}}+2),2^{2^{j}}(2^{2^{j}}+1),2^{2^{j}}+1)$ design having very specific features. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_11407 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Higman-M\lowercase{c}Laughlin Theorem for the flag-transitive $2$-designs with $λ$ prime Montinaro, Alessandro Combinatorics Group Theory A famous result of Higman and McLaughlin \cite{HM} in 1961 asserts that any flag-transitive automorphism group $G$ of a $2$-design $\mathcal{D}$ with $λ=1$ acts point-primitively on $\mathcal{D}$. In this paper, we show that the Higman and McLaughlin theorem is still true when $λ$ is a prime and $\mathcal{D}$ is not isomorphic to one of the two $2$-$(16,6,2)$ designs as in [42, Section 1.2], or the $2$-$(45,12,3)$ design as in [44, Construction 4.2], or, when $2^{2^{j}}+1$ is a Fermat prime, a possible $2$-$(2^{2^{j+1}}(2^{2^{j}}+2),2^{2^{j}}(2^{2^{j}}+1),2^{2^{j}}+1)$ design having very specific features. |
| title | The Higman-M\lowercase{c}Laughlin Theorem for the flag-transitive $2$-designs with $λ$ prime |
| topic | Combinatorics Group Theory |
| url | https://arxiv.org/abs/2504.11407 |