Proof of a~conjecture of Wiegold for nilpotent Lie algebras
Sbornik. Mathematics, Tome 211 (2020) no. 12, pp. 1795-1800
Voir la notice de l'article provenant de la source Math-Net.Ru
Let $\mathfrak{g}$ be a nilpotent Lie algebra. By the breadth $b(x)$ of an element $x$ of $\mathfrak{g}$ we mean the number $[\mathfrak{g}:C_{\mathfrak{g}}(x)]$. Vaughan-Lee showed that if the breadth of all elements of the Lie algebra $\mathfrak{g}$ is bounded by a number $n$, then the dimension of the commutator subalgebra of the Lie algebra does not exceed $n(n+1)/2$. We show that if $\dim \mathfrak{g'} > n(n+1)/2$ for some nonnegative $n$, then the Lie algebra $\mathfrak{g}$ is generated by the elements of breadth $>n$, and thus we prove a conjecture due to Wiegold (Question 4.69 in the Kourovka Notebook) in the case of nilpotent Lie algebras.
Bibliography: 4 titles.
Keywords:
nilpotent Lie algebras, finite $p$-groups, breadth of an element, estimate for the size of the commutator subalgebra.
@article{SM_2020_211_12_a4,
author = {A. A. Skutin},
title = {Proof of a~conjecture of {Wiegold} for nilpotent {Lie} algebras},
journal = {Sbornik. Mathematics},
pages = {1795--1800},
publisher = {mathdoc},
volume = {211},
number = {12},
year = {2020},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_2020_211_12_a4/}
}
A. A. Skutin. Proof of a~conjecture of Wiegold for nilpotent Lie algebras. Sbornik. Mathematics, Tome 211 (2020) no. 12, pp. 1795-1800. http://geodesic.mathdoc.fr/item/SM_2020_211_12_a4/