Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 ${\overline X}^G = \bigcap \{ \ker \varphi | \varphi : X\rightarrow G, \mbox { with } N \subseteq \ker \varphi \}$ 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?)

  • R. Camm, Simple free products, Journ. London Math. Soc. 28 (1953) 66–76.
  • V. D. Mazurov and E. I. Khukhro (eds.), Unsolved problems in group theory. The Kourovka notebook, Thirteenth augmented edition, Russian Academy of Sciences Siberian Division, Institute of Mathematics, Novosibirsk, 1995. MR 1392713
  • B. Plotkin, Radicals in groups, operations on classes of groups, and radical classes, Transl., II Ser. Amer. Math. Soc. 119, (1983) 89–118.
  • B. I. Plotkin, Radicals and verbals, Radical theory (Eger, 1982) Colloq. Math. Soc. János Bolyai, vol. 38, North-Holland, Amsterdam, 1985, pp. 379–403. MR 899121
  • B. Plotkin, Universal algebra, algebraic logic, and databases, Mathematics and its Applications, vol. 272, Kluwer Academic Publishers Group, Dordrecht, 1994. Translated from the 1991 Russian original by J. Cīrulis, A. Nenashev and V. Pototsky and revised by the author. MR 1273136
  • B. Plotkin, E. Plotkin, and A. Tsurkov, Geometrical equivalence of groups, Comm. Algebra 27 (1999), no. 8, 4015–4025. MR 1700201, DOI
  • Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin-New York, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. MR 0577064

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 20E06, 20E10, 20E32, 20F06

Retrieve articles in all journals with MSC (2000): 20E06, 20E10, 20E32, 20F06

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
MR Author ID: 160185
ORCID: 0000-0003-0462-3152

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