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Radicals and Plotkin's problem concerning geometrically equivalent groups

Authors: Rüdiger Göbel and Saharon Shelah
Journal: Proc. Amer. Math. Soc. 130 (2002), 673-674
MSC (2000): Primary 20E06, 20E10, 20E32; Secondary 20F06
Published electronically: September 28, 2001
MathSciNet review: 1866018
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Abstract: 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 ${\overline X}^G = \bigcap \{ \ker \varphi \vert \varphi : X\rightarrow G, \mbox{ with } N \subseteq \ker \varphi \}$ 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

Saharon Shelah
Affiliation: Department of Mathematics, Hebrew University, Jerusalem, Israel–and–Rutgers University, New Brunswick, New Jersey

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
Article copyright: © Copyright 2001 American Mathematical Society