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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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