Abstract:
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.
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