# Sh:741

• Göbel, R., & Shelah, S. (2002). Radicals and Plotkin’s problem concerning geometrically equivalent groups. Proc. Amer. Math. Soc., 130(3), 673–674.
• 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\{{\rm ker}\varphi|\varphi:X\rightarrow G, \text{ with } N \subseteq{\rm 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. Quasiidentities 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 quasiidentiies. Plotkin showed that nilpotent groups G and H satisfy the same quasiidenties 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.
• Version 2000-09-18_11 (3p) published version (2p)
Bib entry
@article{Sh:741,
author = {G{\"o}bel, R{\"u}diger and Shelah, Saharon},
title = {{Radicals and Plotkin's problem concerning geometrically equivalent groups}},
journal = {Proc. Amer. Math. Soc.},
fjournal = {Proceedings of the American Mathematical Society},
volume = {130},
number = {3},
year = {2002},
pages = {673--674},
issn = {0002-9939},
mrnumber = {1866018},
mrclass = {20E06 (20E10)},
doi = {10.1090/S0002-9939-01-06108-1},
note = {\href{https://arxiv.org/abs/math/0010303}{arXiv: math/0010303}},
arxiv_number = {math/0010303}
}