Infinite locally soluble k -Engel groups

Lucia Serena Spiezia

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1992)

  • Volume: 3, Issue: 3, page 177-183
  • ISSN: 1120-6330

Abstract

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In this paper we deal with the class E k * of groups G for which whenever we choose two infinite subsets X , Y there exist two elements x ∈ X , y ∈ Y such that x , y , … , y ︸ k = 1 . We prove that an infinite finitely generated soluble group in the class E k * is in the class E k of k -Engel groups. Furthermore, with k = 2 , we show that if G ∈ E 2 * is infinite locally soluble or hyperabelian group then G ∈ E 2 .

How to cite

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Spiezia, Lucia Serena. "Infinite locally soluble \( k \)-Engel groups." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 3.3 (1992): 177-183. <http://eudml.org/doc/244332>.

@article{Spiezia1992,
abstract = {In this paper we deal with the class \( \mathcal\{E\}\_\{k\}^\{*\} \) of groups \( G \) for which whenever we choose two infinite subsets \( X \), \( Y \) there exist two elements \( x \in X \), \( y \in Y \) such that \( [x, \underbrace\{y,\ldots,y\}\_\{k\}] = 1 \). We prove that an infinite finitely generated soluble group in the class \( \mathcal\{E\}\_\{k\}^\{*\} \) is in the class \( \mathcal\{E\}\_\{k\} \) of \( k \)-Engel groups. Furthermore, with \( k = 2 \), we show that if \( G \in \mathcal\{E\}\_\{2\}^\{*\} \) is infinite locally soluble or hyperabelian group then \( G \in \mathcal\{E\}\_\{2\} \).},
author = {Spiezia, Lucia Serena},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Groups; Engel; Varieties; infinite subsets; finitely generated soluble group; -Engel groups; infinite locally soluble; hyperabelian group},
language = {eng},
month = {9},
number = {3},
pages = {177-183},
publisher = {Accademia Nazionale dei Lincei},
title = {Infinite locally soluble \( k \)-Engel groups},
url = {http://eudml.org/doc/244332},
volume = {3},
year = {1992},
}

TY - JOUR
AU - Spiezia, Lucia Serena
TI - Infinite locally soluble \( k \)-Engel groups
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1992/9//
PB - Accademia Nazionale dei Lincei
VL - 3
IS - 3
SP - 177
EP - 183
AB - In this paper we deal with the class \( \mathcal{E}_{k}^{*} \) of groups \( G \) for which whenever we choose two infinite subsets \( X \), \( Y \) there exist two elements \( x \in X \), \( y \in Y \) such that \( [x, \underbrace{y,\ldots,y}_{k}] = 1 \). We prove that an infinite finitely generated soluble group in the class \( \mathcal{E}_{k}^{*} \) is in the class \( \mathcal{E}_{k} \) of \( k \)-Engel groups. Furthermore, with \( k = 2 \), we show that if \( G \in \mathcal{E}_{2}^{*} \) is infinite locally soluble or hyperabelian group then \( G \in \mathcal{E}_{2} \).
LA - eng
KW - Groups; Engel; Varieties; infinite subsets; finitely generated soluble group; -Engel groups; infinite locally soluble; hyperabelian group
UR - http://eudml.org/doc/244332
ER -

References

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  1. GRUENBERG, K. W., The upper central series in soluble groups. Illinois J. of Math., 5, 1961, 436-466. Zbl0244.20028MR136657
  2. LONGOBARDI, P. - MAJ, M. - RHEMTULLA, A. H., Infinite groups in a given variety and Ramsey's theorem. Communications in Algebra, to appear. Zbl0751.20020
  3. KIM, P. S. - RHEMTULLA, A. H. - SMITH, H., A characterization of infinite metabelian groups. Houston J. of Math., to appear. Zbl0744.20033
  4. ROBINSON, D. J. S., Finiteness Conditions and Generalized Soluble Groups. Part I and Part II. Springer Verlag, Berlin-Heidelberg-New York1972. Zbl0243.20033
  5. ZAICEV, D. I., On solvable subgroups of locally solvable groups. Dokl. Akad. Nauk SSSR, 214, 1974, 1250-1253 (translation in Soviet Math. Dokl., 15, 1974, 342-345). Zbl0322.20017MR338181

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