Radicals and Plotkin's problem concerning geometrically equivalent groups |
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Authors: | Rü diger Gö bel Saharon Shelah |
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Institution: | Fachbereich 6, Mathematik und Informatik, Universität Essen, 45117 Essen, Germany ; Department of Mathematics, Hebrew University, Jerusalem, Israel--and--Rutgers University, New Brunswick, New Jersey |
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Abstract: | If and are groups and is a normal subgroup of , then the -closure of in is the normal subgroup of . In particular, is the -radical of . Plotkin calls two groups and geometrically equivalent, written , if for any free group of finite rank and any normal subgroup of the -closure and the -closure of in are the same. Quasi-identities are formulas of the form for any words in a free group. Generally geometrically equivalent groups satisfy the same quasi-identities. Plotkin showed that nilpotent groups and satisfy the same quasi-identities if and only if and are geometrically equivalent. Hence he conjectured that this might hold for any pair of groups. We provide a counterexample. |
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Keywords: | |
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