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Proof of a conjecture of Wiegold for nilpotent Lie algebras
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Abstract
Let
be a nilpotent Lie algebra. By the breadth
of an element
of
we mean the number
. Vaughan-Lee showed that if the breadth of all elements of the Lie algebra
is bounded by a number
, then the dimension of the commutator subalgebra of the Lie algebra does not exceed
. We show that if
for some nonnegative
, then the Lie algebra
is generated by the elements of breadth
, 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.
Title: Proof of a conjecture of Wiegold for nilpotent Lie algebras
Description:
Abstract
Let
be a nilpotent Lie algebra.
By the breadth
of an element
of
we mean the number
.
Vaughan-Lee showed that if the breadth of all elements of the Lie algebra
is bounded by a number
, then the dimension of the commutator subalgebra of the Lie algebra does not exceed
.
We show that if
for some nonnegative
, then the Lie algebra
is generated by the elements of breadth
, 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.
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