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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.

References [Enhancements On Off] (What's this?)

  • 1. R. Camm, Simple free products, Journ. London Math. Soc. 28 (1953) 66-76. MR 14:616f
  • 2. E.I. Khukhro and V.D. Mazurov, Unsolved problems in group theory; the Kourovka Notebook, Russian Academy of Science, Novosibirsk, 1999. 13th ed. 1995 MR 97d:20001
  • 3. B. Plotkin, Radicals in groups, operations on classes of groups, and radical classes, Transl., II Ser. Amer. Math. Soc. 119, (1983) 89-118.
  • 4. B. Plotkin, Radicals and verbals, Radical theory, Colloqu. Math. Soc. Janos Bolyai 38, (1985) 379-403. MR 88f:16008
  • 5. B. Plotkin, Universal Algebra, Algebraic Logic, and Databases, Kluwer Acad. Publ. Dordrecht, Boston, London 1994. MR 95c:68061
  • 6. B. Plotkin, E. Plotkin, A. Tsurkov, Geometrical equivalence of groups, Commun. Algebra 27, (1999) 4015-4025. MR 2000e:08006
  • 7. R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer Ergebinsberichte 89, Berlin-Heidelberg-New York, 1977. MR 58:28182

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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

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