{"paper":{"title":"Partial Group Symmetry in Figures I: Semidirect Products and the Six Coins","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"math.GR","authors_text":"Takahiro Hayashi","submitted_at":"2025-06-17T08:30:45Z","abstract_excerpt":"In this paper, we construct a partial group \\(\\mathcal{P}(F)\\) that represents the \"partial symmetry\" inherent in a subset \\(F\\) of \\(d\\)-dimensional Euclidean space. In cases where \\(F\\) is not connected, \\(\\mathcal{P}(F)\\) captures more detailed information than the conventional symmetry group \\(G(F)\\). To establish a stronger connection between \\(\\mathcal{P}(F)\\) and \\(F\\), we introduce a novel definition of partial group action. Furthermore, to characterize \\(\\mathcal{P}(F)\\) in specific cases, we define partial group actions on other partial groups and present a construction of the corres"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.14304","kind":"arxiv","version":2},"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/2506.14304/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"}