{"paper":{"title":"The Higman-M\\lowercase{c}Laughlin Theorem for the flag-transitive $2$-designs with $\\lambda$ prime","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"Alessandro Montinaro","submitted_at":"2025-04-15T17:23:58Z","abstract_excerpt":"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 $\\lambda=1$ acts point-primitively on $\\mathcal{D}$. In this paper, we show that the Higman and McLaughlin theorem is still true when $\\lambda$ 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 havin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.11407","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2504.11407/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}