Left 3-Engel elements of odd order in groups

Gunnar Traustason, Enrico Jabara, Pham Huu Tiep

Research output: Contribution to journalArticle

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.
LanguageEnglish
Pages1921-1927
JournalProceedings of the American Mathematical Society
Volume147
Issue number5
Early online date18 Jan 2019
DOIs
StatusE-pub ahead of print - 18 Jan 2019

Cite this

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

In: Proceedings of the American Mathematical Society, Vol. 147, No. 5, 18.01.2019, p. 1921-1927.

Research output: Contribution to journalArticle

Traustason, Gunnar ; Jabara, Enrico ; Huu Tiep, Pham. / Left 3-Engel elements of odd order in groups. In: Proceedings of the American Mathematical Society. 2019 ; Vol. 147, No. 5. pp. 1921-1927.
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