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Given any $n\\geq 1$, the subgroup $K_n=\\{x+a_{n+1}x^{n+1}+a_{n+2}x^{n+2}+\\cdots\\,|\\, a_i\\in R\\}$ is normal in $G$, and the quotient $G_n=G/K_n$ is the group of truncated polynomials over $R$ of degree $\\leq n$ under substitution. In this paper, we compute the exponent of the image of $K_r$ in $G_n$, for all $r,n\\geq 1$, indicating in every case a family of elements realizing this exponent.","authors_text":"Agust\\'in D'Alessandro, Fernando Szechtman","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-06-09T21:31:04Z","title":"Substitution groups of formal power series"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11461","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:cfa6d30736bd17c77957d94e43fa75b93105f07d1541f338160a8fd6f29a2ca2","target":"record","created_at":"2026-06-11T01:09:49Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"dc69cf841dbef45ad66fcba5018f64eeb6a712cc44cfa683315413a6aa5797a5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-06-09T21:31:04Z","title_canon_sha256":"6c8f618a6f4dd79a65e647261864ca4a7e319382cdfe0b0f453695323cde22db"},"schema_version":"1.0","source":{"id":"2606.11461","kind":"arxiv","version":1}},"canonical_sha256":"58b4eb7d93dd725f51128aa0066ce9a2097bb1fef0cc67cbbd954b680df78151","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"58b4eb7d93dd725f51128aa0066ce9a2097bb1fef0cc67cbbd954b680df78151","first_computed_at":"2026-06-11T01:09:49.813377Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-11T01:09:49.813377Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"356VZugbxwuk/eMddKzv79rZtoO/ntZF6g/OnKdcGXieBC5xsxKki6HIbA0ShIvsDmqYcuIkS5IKdYWE5V90Dw==","signature_status":"signed_v1","signed_at":"2026-06-11T01:09:49.814181Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.11461","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cfa6d30736bd17c77957d94e43fa75b93105f07d1541f338160a8fd6f29a2ca2","sha256:2adc9dd544b8685a1469dd6063d40ceaa367035545dfe64666cb21a5baca0be0"],"state_sha256":"b3b8cedc57f1758d6bf6f382edc0567401b3c52e2147ba9d3a3a33c12887996b"}