### 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 language | English (US) |
---|---|

Pages (from-to) | 1921-1927 |

Number of pages | 7 |

Journal | Proceedings of the American Mathematical Society |

Volume | 147 |

Issue number | 5 |

DOIs | |

State | Published - May 1 2019 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*147*(5), 1921-1927. https://doi.org/10.1090/proc/14389

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

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 147, no. 5, pp. 1921-1927. https://doi.org/10.1090/proc/14389

}

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 -