Proof of a conjecture of Wiegold for nilpotent Lie algebras
Sbornik. Mathematics, Tome 211 (2020) no. 12, pp. 1795-1800
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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},
year = {2020},
volume = {211},
number = {12},
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/
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[3] M. R. Vaughan-Lee, “Breadth and commutator subgroups of $p$-groups”, J. Algebra, 32:2 (1974), 278–285 | DOI | MR | Zbl
[4] A. Skutin, “Proof of a conjecture of Wiegold”, J. Algebra, 526 (2019), 1–5 | DOI | MR | Zbl