{"paper":{"title":"On asymorphisms of groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Igor Protasov, Serhii Slobodianiuk","submitted_at":"2016-02-27T09:47:56Z","abstract_excerpt":"Let $G$, $H$ be groups and $\\kappa$ be a cardinal. A bijection $f:G\\to H$ is caled on asymorphism if, for any $X\\in[G]^{<\\kappa}$, $Y\\in[H]^{<\\kappa}$, there exist $X'\\in[G]^{<\\kappa}$, $Y'\\in[H]^{<\\kappa}$ such that for all $x\\in G$ and $y\\in H$, we have $f(Xx)\\subseteq Y'f(x)$, $f^{-1}(Yy)\\subseteq X'f^{-1}(y)$. For a set $S$, $[S]^{<\\kappa}$ denotes the set $\\{S'\\subseteq S: |S'|<\\kappa\\}$.\n  Let $\\kappa$ and $\\gamma$ be cardinals such that $\\aleph_0<\\kappa\\le\\gamma$. We prove that any two Abelian groups of cardinality $\\gamma$ are $\\kappa$-asymorphic, but the free group of rank $\\gamma$ is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.08577","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"}