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In the particular case when $|g_1|, \\ldots, |g_d|$ can be chosen to be prime-powers we say that $G$ is prime-power coprimely-invariably generated. We will discuss these properties, proving also that the second one is stronger than the first, but that in the particular case of finite soluble groups they"},"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":"1409.0997","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-09-03T08:58:51Z","cross_cats_sorted":[],"title_canon_sha256":"fcd20cd373318d62830d2c5bfaab5efc642f17f2086ca3c5e4ad055b0cb2a300","abstract_canon_sha256":"290d5e4e4f376e57fa450ad22775716a8b8eea8c73dc677761aff03a3384bb17"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:43:44.597773Z","signature_b64":"2z1YI7nyPmWasc01riELMFxDgPgZCRX8JibtOZHduYBQPFVxlL6UZvAMlkQSOeKfp0GM3bE4Wr+8aPIDBfqVDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a67260e602935e3785cce2f7d80cd96d114c56988072466e80b1a99e2656a713","last_reissued_at":"2026-05-18T02:43:44.597364Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:43:44.597364Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Invariable generation with elements of coprime prime-power order","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Andrea Lucchini, Eloisa Detomi","submitted_at":"2014-09-03T08:58:51Z","abstract_excerpt":"A finite group $G$ is coprimely-invariably generated if there exists a set of generators $\\{g_1, \\ldots, g_d\\}$ of $G$ with the property that the orders $|g_1|, \\ldots, |g_d|$ are pairwise coprime and that for all $x_1, \\ldots, x_d \\in G$ the set $\\{g_1^{x_1}, \\ldots, g_d^{x_d}\\}$ generates $G$. In the particular case when $|g_1|, \\ldots, |g_d|$ can be chosen to be prime-powers we say that $G$ is prime-power coprimely-invariably generated. 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