### 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) |
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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 2019 |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Proceedings of the American Mathematical Society*,

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