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Let $d(G)$ denote the minimal non negative integer for which there exists a subset $X$ of $G$ with $d(G)$ elements such that the normal subgroup of $G$ generated by $X$ is all of $G$. It is known that $d(G)\\leq ram^{t}(G)$. However, it is unknown whether or not every finite group $G$ can be realized as a Galois group of a tamely ramified extension of $\\mathbb{Q}$ with exactly $"},"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":"1611.04103","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-11-13T08:44:40Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"8536e4f4ad19725027e99e8c86cd8288a4b52ffee1c8418d1b6722f0f846370b","abstract_canon_sha256":"5572239e1279777e0812837dc4566cd3930b76f2f1dba861705006714348fef7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:59:17.552687Z","signature_b64":"5jJhhc3t1q5JMZ2uiwDaTcub6gqjysVq8EBt1+1xvb483OCDUse7oUc2YEr0vr9cACF12QFw8SETCyBzyzeICg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6e3d42a853a9867966bd2aea57e1f9e2b9d232e9b3447813b2bece87749206c7","last_reissued_at":"2026-05-18T00:59:17.551969Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:59:17.551969Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Upper bound on the number of ramified primes for odd order solvable groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.NT","authors_text":"Daniel Rabayev","submitted_at":"2016-11-13T08:44:40Z","abstract_excerpt":"Let $G$ be a finite group and let $ram^{t}(G)$ denote the minimal positive integer $n$ such that $G$ can be realized as the Galois group of a tamely ramified extension of $\\mathbb{Q}$ ramified only at $n$ finite primes. Let $d(G)$ denote the minimal non negative integer for which there exists a subset $X$ of $G$ with $d(G)$ elements such that the normal subgroup of $G$ generated by $X$ is all of $G$. It is known that $d(G)\\leq ram^{t}(G)$. 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