### Abstract

It is still an open problem to determine whether the n-th
Engel word [x,_n y] is concise, that is, if for every group G such that
the set of values e_n(G) taken by [x,_n y] on G is finite it follows that
the verbal subgroup E_n(G) generated by e_n(G) is also finite. We prove
that if e_n(G) is finite then [En_(G),G] is finite, and either G=[E_n(G),G]
is locally nilpotent and E_n(G) is finite, or G has a finitely generated
section that is an infinite simple n-Engel group. It follows that [x_n y]
is concise if n is at most four.

Original language | English |
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Pages (from-to) | 2570-2576 |

Journal | Communications in Algebra |

Volume | 40 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2012 |

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## Cite this

Fernandez-Alcober, G. A., Moriagi, M., & Traustason, G. (2012). A note on conciseness of Engel words.

*Communications in Algebra*,*40*(7), 2570-2576. https://doi.org/10.1080/00927872.2011.582061