Abstract
We determine the number of centralizers of different non-abelian finite dimensional Lie algebras over a specific field. Also, the concept of Lie algebras with abelian centralizers are studied and using a result of Bokut and Kukin [5], for a given residually free Lie algebra L, it is shown that L is fully residually free if and only if every centralizer of non-zero elements of L is abelian.
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A. Abdollahi, S. M. J. Amiri and A. M. Hassanabadi, Groups with specific number of centralizers, Houston J. Math., 33(1) (2007), 43–57.
A. R. Ashrafi, On finite groups with a given number of centralizers, Algebra Colloq., 7(2) (2000), 139–146.
B. Baumslag, Residually free groups, Proc. London Math. Soc., 17(3) (1967), 402–418.
S. M. Belcastro and G. J. Sherman, Counting centralizers in finite groups, Math. Magazine, 67(5) (1994), 366–374.
L. A. Bokut and G. P. Kukin, Algorithmic and combinatorial algebra, Kluwer Academic Publisher, 1994.
L. Ciobanu, B. Fine and G. Rosenberger, Classes of groups generalizing a theorem of Benjamin Baumslag, Comm. Algebra, 44(2) (2016), 656–667.
C. Delizia, P. Moravec and C. Nicotera, Finite groups in which some property of two-generator subgroups is transitive, Bull. Aust. Math. Soc., 75 (2007), 313–320.
Karin Erdmann and Mark J. Wildon. Introduction to Lie algebras, Undergraduate Mathematics Series. Springer-Verlag, London Ltd., 2006.
I. Klep and P. Moravec, Lie algebras with abelian centralizers, Algebra Colloq., 17(4) (2010), 629–636.
E. Marshall, The Frattini subalgebra of a Lie algebra, J. Lond. Math. Soc., 42 (1967), 416–422.
L. Weisner, Groups in which the normaliser of every element except identity is abelian, Bull. Amer. Math. Soc., 31 (1925), 413–416.
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Saffarnia, S., Moghaddam, M.R.R. & Rostamyari, M.A. Centralizers in Lie Algebras. Indian J Pure Appl Math 49, 39–49 (2018). https://doi.org/10.1007/s13226-018-0252-0
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DOI: https://doi.org/10.1007/s13226-018-0252-0