{"paper":{"title":"Absolutely split metacyclic groups and weak metacirculants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jin-Xin Zhou, Li Cui","submitted_at":"2018-01-26T05:51:03Z","abstract_excerpt":"Let $m,n,r$ be positive integers, and let $G=\\langle a\\rangle: \\langle b\\rangle \\cong \\mathbb{Z}_n: \\mathbb{Z}_m$ be a split metacyclic group such that $b^{-1}ab=a^r$. We say that $G$ is {\\em absolutely split with respect to $\\langle a\\rangle$} provided that for any $x\\in G$, if $\\langle x\\rangle\\cap\\langle a\\rangle=1$, then there exists $y\\in G$ such that $x\\in\\langle y\\rangle$ and $G=\\langle a\\rangle: \\langle y\\rangle$. In this paper, we give a sufficient and necessary condition for the group $G$ being absolutely split. This generalizes a result of Sanming Zhou and the second author in [arXi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.08681","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}