# 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.
• Current version: 2000-09-18_10 (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}
}
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9148724675178528, "perplexity": 3797.641780521166}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400244231.61/warc/CC-MAIN-20200926134026-20200926164026-00530.warc.gz"}