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We show that if $ gcd(m-1,q)$ is a divisor of $r$ then there are exactly $\\varphi(q). gcd(m-1,q)$ conjugacy classes of elements of order $q$ in $T_{r,m}$, where $\\varphi$ is the Euler function phi. As a corollary, we obtain that the Thompson group $T$ is isomorphic to none of the groups $T_{r,m}$, for $m\\not=2$ and any morphism from"},"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":"1809.08584","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-09-23T12:11:55Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"baf73b23bda054e3d14587379c1aa38eec6112b58aac820e5d23981ec21de1e8","abstract_canon_sha256":"4e9a9006e6dc47ef1a363333a9ff78f7106cff0e06913d08ece531ad65396fa2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:04.033441Z","signature_b64":"/oOXDiitFceHoh7OldgSCHpxlHoUrb2rQHHrIEterC59UoMff6BO21Ajylut42orAYFUkDEYBN2f81ikTjwiBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4557b0efa83917ba792b064f997e5d1f0ec85055cf180510f7b26a60f0ed45d0","last_reissued_at":"2026-05-18T00:05:04.032794Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:04.032794Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Nombre de classes de conjugaison d'\\'el\\'ements d'ordre fini dans les groupes de Brown-Thompson","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.GR","authors_text":"Hajer Hmili, Isabelle Liousse","submitted_at":"2018-09-23T12:11:55Z","abstract_excerpt":"We extend a result of Matucci on the number of conjugacy classes of finite order elements in the Thompson group $T$. According to Liousse, if $ gcd(m-1,q)$ is not a divisor of $r$ then there does not exist element of order $q$ in the Brown-Thompson group $T_{r,m}$. We show that if $ gcd(m-1,q)$ is a divisor of $r$ then there are exactly $\\varphi(q). gcd(m-1,q)$ conjugacy classes of elements of order $q$ in $T_{r,m}$, where $\\varphi$ is the Euler function phi. 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