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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Radicals and Plotkin’s problem concerning geometrically equivalent groups
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by Rüdiger Göbel and Saharon Shelah PDF
Proc. Amer. Math. Soc. 130 (2002), 673-674 Request permission


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 ${\overline X}^G = \bigcap \{ \ker \varphi | \varphi : X\rightarrow G, \mbox { with } N \subseteq \ker \varphi \}$ of $X$. In particular, ${\overline 1}^G = R_GX$ is the $G$-radical of $X$. Plotkin calls two groups $G$ and $H$ geometrically equivalent, written $G\sim H$, 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. Quasi-identities are formulas of the form $(\bigwedge _{i\le n} w_i = 1 \rightarrow w =1)$ for any words $w, w_i \ (i\le n)$ in a free group. Generally geometrically equivalent groups satisfy the same quasi-identities. Plotkin showed that nilpotent groups $G$ and $H$ satisfy the same quasi-identities if and only if $G$ and $H$ are geometrically equivalent. Hence he conjectured that this might hold for any pair of groups. We provide a counterexample.
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Additional Information
  • Rüdiger Göbel
  • Affiliation: Fachbereich 6, Mathematik und Informatik, Universität Essen, 45117 Essen, Germany
  • Email:
  • Saharon Shelah
  • Affiliation: Department of Mathematics, Hebrew University, Jerusalem, Israel–and–Rutgers University, New Brunswick, New Jersey
  • MR Author ID: 160185
  • ORCID: 0000-0003-0462-3152
  • Email:
  • Received by editor(s): September 6, 2000
  • Received by editor(s) in revised form: September 21, 2000
  • Published electronically: September 28, 2001
  • Additional Notes: The authors were supported by project No. G 0545-173, 06/97 of the German-Israeli Foundation for Scientific Research & Development. This paper is #GbSh 741 in Shelah’s list of publications.
  • Communicated by: Stephen D. Smith
  • © Copyright 2001 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 130 (2002), 673-674
  • MSC (2000): Primary 20E06, 20E10, 20E32; Secondary 20F06
  • DOI:
  • MathSciNet review: 1866018