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On the other hand, $3$-transposition groups $(G,D)$ can be algebraically characterised as Matsuo algebras $M_\\alpha(G,D)$ with idempotents satisfying the fusion rules $\\Phi(\\alpha)$ for some $\\alpha$. We classify the Jordan algebras $J$ which are isomorphic to a Matsuo algebra $M_{1/2}(G,D)$, in which case $(G,D)$ is a subgroup of the (algebraic) automorphism group of $J$; the only possibilities are $G = \\operatorname{S"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1502.05657","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-02-19T18:04:36Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"23f663c54d39635b5c5766152423a1a7628ab0e37922c711f156cfa524b60252","abstract_canon_sha256":"18c8aaf0ff1625806be2326ffca7f81c367eb39e7d5c5d362f26f486c7e4e7a3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:31:03.267839Z","signature_b64":"hyF7a46wtCtYRE3sC33eM6qb6A+PtwJA9nJeZ5vy+NS0fi5O2BPXZnaFKQImxBhcAiFaNknTquYecmuJ8sVNDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"abe7fcab4a72c2e7a70c4c4af85dce7c487eb06b483143b7a759576e6be8cf7f","last_reissued_at":"2026-05-18T01:31:03.267350Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:31:03.267350Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Jordan algebras and 3-transposition groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.RA","authors_text":"Felix Rehren, Tom De Medts","submitted_at":"2015-02-19T18:04:36Z","abstract_excerpt":"An idempotent in a Jordan algebra induces a Peirce decomposition of the algebra into subspaces whose pairwise multiplication satisfies certain fusion rules $\\Phi(\\frac{1}{2})$. 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