Left 3-engel elements of odd order in groups

Enrico Jabara, Gunnar Traustason, Pham Huu Tiep

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Let G be a group and let x ε G be a left 3-Engel element of odd order. We show that x is in the locally nilpotent radical of G. We establish this by proving that any finitely generated sandwich group, generated by elements of odd orders, is nilpotent. This can be seen as a group theoretic analog of a well-known theorem on sandwich algebras by Kostrikin and Zel’manov. We also give some applications of our main result. In particular, for any given word w = w(x1, . . ., xn) in n variables, we show that if the variety of groups satisfying the law w 3 = 1 is a locally finite variety of groups of exponent 9, then the same is true for the variety of groups satisfying the law (x 3 n +1 w 3 ) 3 = 1.

Original languageEnglish (US)
Pages (from-to)1921-1927
Number of pages7
JournalProceedings of the American Mathematical Society
Volume147
Issue number5
DOIs
StatePublished - May 1 2019

Fingerprint

Algebra
Odd
Sandwich
Finitely Generated
Exponent
Analogue
Theorem

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Left 3-engel elements of odd order in groups. / Jabara, Enrico; Traustason, Gunnar; Tiep, Pham Huu.

In: Proceedings of the American Mathematical Society, Vol. 147, No. 5, 01.05.2019, p. 1921-1927.

Research output: Contribution to journalArticle

Jabara, Enrico ; Traustason, Gunnar ; Tiep, Pham Huu. / Left 3-engel elements of odd order in groups. In: Proceedings of the American Mathematical Society. 2019 ; Vol. 147, No. 5. pp. 1921-1927.
@article{c712faf90ea74358b61ffe2ccd3ef9ca,
title = "Left 3-engel elements of odd order in groups",
abstract = "Let G be a group and let x ε G be a left 3-Engel element of odd order. We show that x is in the locally nilpotent radical of G. We establish this by proving that any finitely generated sandwich group, generated by elements of odd orders, is nilpotent. This can be seen as a group theoretic analog of a well-known theorem on sandwich algebras by Kostrikin and Zel’manov. We also give some applications of our main result. In particular, for any given word w = w(x1, . . ., xn) in n variables, we show that if the variety of groups satisfying the law w 3 = 1 is a locally finite variety of groups of exponent 9, then the same is true for the variety of groups satisfying the law (x 3 n +1 w 3 ) 3 = 1.",
author = "Enrico Jabara and Gunnar Traustason and Tiep, {Pham Huu}",
year = "2019",
month = "5",
day = "1",
doi = "10.1090/proc/14389",
language = "English (US)",
volume = "147",
pages = "1921--1927",
journal = "Proceedings of the American Mathematical Society",
issn = "0002-9939",
publisher = "American Mathematical Society",
number = "5",

}

TY - JOUR

T1 - Left 3-engel elements of odd order in groups

AU - Jabara, Enrico

AU - Traustason, Gunnar

AU - Tiep, Pham Huu

PY - 2019/5/1

Y1 - 2019/5/1

N2 - Let G be a group and let x ε G be a left 3-Engel element of odd order. We show that x is in the locally nilpotent radical of G. We establish this by proving that any finitely generated sandwich group, generated by elements of odd orders, is nilpotent. This can be seen as a group theoretic analog of a well-known theorem on sandwich algebras by Kostrikin and Zel’manov. We also give some applications of our main result. In particular, for any given word w = w(x1, . . ., xn) in n variables, we show that if the variety of groups satisfying the law w 3 = 1 is a locally finite variety of groups of exponent 9, then the same is true for the variety of groups satisfying the law (x 3 n +1 w 3 ) 3 = 1.

AB - Let G be a group and let x ε G be a left 3-Engel element of odd order. We show that x is in the locally nilpotent radical of G. We establish this by proving that any finitely generated sandwich group, generated by elements of odd orders, is nilpotent. This can be seen as a group theoretic analog of a well-known theorem on sandwich algebras by Kostrikin and Zel’manov. We also give some applications of our main result. In particular, for any given word w = w(x1, . . ., xn) in n variables, we show that if the variety of groups satisfying the law w 3 = 1 is a locally finite variety of groups of exponent 9, then the same is true for the variety of groups satisfying the law (x 3 n +1 w 3 ) 3 = 1.

UR - http://www.scopus.com/inward/record.url?scp=85065464915&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85065464915&partnerID=8YFLogxK

U2 - 10.1090/proc/14389

DO - 10.1090/proc/14389

M3 - Article

AN - SCOPUS:85065464915

VL - 147

SP - 1921

EP - 1927

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 5

ER -