On dimension of Lie Algebras and nilpotent Lie algebras

Document Type : Original Article

Author

Department of mathematics, Islamic Azad university, Neyshabour branch, Mashhad, Iran

Abstract

Schur proved that if the center of a group G has finite index, then the derived subgroup G′ is also finite. Moneyhun proved that if L is a Lie algebra such that dim(L/Z(L)) = n, then dim(L^2) ≤1/2n(n-1) In this paper, we extend the converse of Moneyhun’s theorem
. Also, we prove a wellknown result of nilpotent Lie algebras by using a different technique

Keywords


[1] H. Arabyani, On Dimension of Derived Algebra and the Higher Schur Multiplier of Lie Alge-
bras, Southeast Asian Bull. Math, 45(1) 2021, 1-9.
[2] H. Arabyani, The commutator subgroup of a pair of groups, Tbilisi Mathematical Journal,
14(3) 2021, 171-177.
[3] H. Arabyani, F. Panbehkar, H. Safa, On the structure of factor Lie algebras, Bull. Korean
Math. Soc., 54(2) 2017, 455-461.
[4] H. Arabyani, F. Saeedi, On dimension of derived algebra and central factor of a Lie algebra,
Bull. Iranian Math. Soc, 41(5) 2015, 1093-1102.
[5] C.Y. Chao, Some characterizations of nilpotent Lie algebras, Math. Zcitschr, 103(1) 1968,
40-42.
[6] K. Erdmann, M.J. Wildon, Introduction to Lie Algebras, Springer undergraduate Mathematics
series, 2006.
[7] A. Faramarzi Salles, The Converse of Baer's Theorem, ArXiv:1103.2600v1 [math. GR] 14 Mar
2011.
[8] R. Hatamian, M. Hassanzadeh, S. Kayvanfar, A converse of Baer's theorem, Ricerch Math,
36(1) 2013, 183-187.
[9] I.D. Macdonald, Some explicit bounds in groups with  nite derived groups, Proc. London
Math. Soc., 3(11) 1961, 23-56.
[10] K. Moneyhun, Isoclinism in Lie algebras, Algebras Groups Geom, 11(1) 1994, 9-22.
[11] E.I. Marshall, The Frattini subalgebra of a Lie algebra, J. London Math. Soc., 1(1) 1967,
416-422.
V.V. Morozov, Classi cation des algebras de Lie nilpotents de dimension 6, Izv. Vyssh. Ucheb.
Zar, 4 1958, 161-171.
[12] B. H. Neumann, Groups with  nite classes of conjugate elements, Proc. London Math. Soc.,
3(1) 1951, 178-187.
[13] P. Niroomand, The converse of Schur's theorem, Arch. Math. (Basel), 94(5) 2010, 401-403.
[14] K. Podoski, B. Szegedy, Bounds for the index of the centre in capable groups, Proc. Amer.
Math. Soc., 133(12) 2005, 3441-3445.
[15] D.J.S. Robinson, A course in the theory of groups, Springer-Verlag, New York, 1982.
[16] I. Schur, U
ber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen,
J. Reine Angew. Math., 127 1904, 20-50.
[17] J. Wiegold, Multiplicators and groups with  nite central factor-groups, Math. Z., 89(4) 1965,
345-347.
Volume 3, Issue 1
February 2022
Pages 33-36
  • Receive Date: 23 November 2021
  • Revise Date: 18 December 2021
  • Accept Date: 07 January 2022
  • First Publish Date: 13 January 2022