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Roney-Dougal, Eloisa Detomi","submitted_at":"2014-10-28T10:09:26Z","abstract_excerpt":"A finite group $G$ is \\emph{coprimely-invariably generated} if there exists a set of generators $\\{g_1, ..., g_u\\}$ of $G$ with the property that the orders $|g_1|, ..., |g_u|$ are pairwise coprime and that for all $x_1, ..., x_u \\in G$ the set $\\{g_1^{x_1}, ..., g_u^{x_u}\\}$ generates $G$.\n  We show that if $G$ is coprimely-invariably generated, then $G$ can be generated with three elements, or two if $G$ is soluble, and that $G$ has zero presentation rank. As a corollary, we show that if $G$ is any finite group such that no proper subgroup has the same exponent as $G$, then $G$ has zero pres"},"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":"1410.7569","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-10-28T10:09:26Z","cross_cats_sorted":[],"title_canon_sha256":"869f47997b2bf21aca0f45d234731477eda2abc679ef880d17bc94aa591ec1b9","abstract_canon_sha256":"a6c98c814cb17d984f75b815a586b0f2a911e598fbdcef15b8e41f66820e2b57"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:39:10.003518Z","signature_b64":"TtjJDOL6RTynuGOKZwJt+05NB1rPfW4X9VhU7GZ3ZH/iKaMz38Dza1xnaXKZYgxQXzoaDTuXs6hMyb5Dv28ZAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aad8351815fd97d39318892fdfe93c81d1bdf57e33acacc4d7cf12c50ae6459a","last_reissued_at":"2026-05-18T02:39:10.002787Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:39:10.002787Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Coprime invariable generation and minimal-exponent groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Andrea Lucchini, Colva M. Roney-Dougal, Eloisa Detomi","submitted_at":"2014-10-28T10:09:26Z","abstract_excerpt":"A finite group $G$ is \\emph{coprimely-invariably generated} if there exists a set of generators $\\{g_1, ..., g_u\\}$ of $G$ with the property that the orders $|g_1|, ..., |g_u|$ are pairwise coprime and that for all $x_1, ..., x_u \\in G$ the set $\\{g_1^{x_1}, ..., g_u^{x_u}\\}$ generates $G$.\n  We show that if $G$ is coprimely-invariably generated, then $G$ can be generated with three elements, or two if $G$ is soluble, and that $G$ has zero presentation rank. 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