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

DOI:
https://doi.org/10.1090/S0002-9939-01-06108-1

Published electronically:
September 28, 2001

MathSciNet review:
1866018

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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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Additional Information

**Rüdiger Göbel**

Affiliation:
Fachbereich 6, Mathematik und Informatik, Universität Essen, 45117 Essen, Germany

Email:
R.Goebel@uni-essen.de

**Saharon Shelah**

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

Email:
Shelah@math.huji.ac.il

DOI:
https://doi.org/10.1090/S0002-9939-01-06108-1

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