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On finite groups containing an element whose Engel sink is small
Abstract
For an element g of a group G, a right Engel sink of g is a subset of G containing all sufficiently long commutators [...[[g ,x],x],… ,x] for all x∈ G. A left Engel sink of g is a subset of G containing all sufficiently long commutators [...[[x ,g ],g ],… ,g] for all x∈ G. Using the classification of finite simple groups we prove that if a finite group G has an element g such that G=[G,g], then the order of G is bounded in terms of a right Engel sink of g, as well as in terms of a left Engel sink of g. Earlier Guralnick and Tracey proved this in the case where g is an involution without using the classification.
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BibTeX
@misc{khukhro2026finite,
title = {{On finite groups containing an element whose Engel sink is small}},
author = {Evgeny Khukhro and Pavel Shumyatsky},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2605.08607},
eprint = {2605.08607},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.08607},
}
RIS
TY - UNPB
TI - On finite groups containing an element whose Engel sink is small
AU - Evgeny Khukhro
AU - Pavel Shumyatsky
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2605.08607
UR - https://arxiv.org/abs/2605.08607
AB - For an element g of a group G, a right Engel sink of g is a subset of G containing all sufficiently long commutators [...[[g ,x],x],… ,x] for all x∈ G. A left Engel sink of g is a subset of G containing all sufficiently long commutators [...[[x ,g ],g ],… ,g] for all x∈ G. Using the classification of finite simple groups we prove that if a finite group G has an element g such that G=[G,g], then the order of G is bounded in terms of a right Engel sink of g, as well as in terms of a left Engel sink of g. Earlier Guralnick and Tracey proved this in the case where g is an involution without using the classification.
ER -