{"paper":{"title":"Radicals and Plotkin's problem concerning geometrically equivalent groups","license":"","headline":"","cross_cats":["math.LO"],"primary_cat":"math.GR","authors_text":"R\\\"udiger G\\\"obel, Saharon Shelah","submitted_at":"2000-10-30T19:03:52Z","abstract_excerpt":"If G and X are groups and N is a normal subgroup of X, then the G-closure of N in X is the normal subgroup X^G= bigcap{kerphi|phi:X-> G, with N subseteq kerphi} of X . In particular, 1^G = R_G X is the G-radical of X. Plotkin calls two groups G and H geometrically equivalent, written G H, if for any free group F of finite rank and any normal subgroup N of F the G-closure and the H-closure of N in F are the same. Quasiidentities are formulas of the form (bigwedge_{i<=n}w_i=1 -> w =1) for any words w, w_i (i<=n) in a free group. Generally geometrically equivalent groups satisfy the same quasiide"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0010303","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"}