Lie algebras associated with Sylow $p$-subgroups of finite symmetric groups (in Russian)

V.I.Sushchansky; N.V.Netreba

We study Lie algebras associated with Sylow p-subgroups of symmetric group of degree $p^n$ (p is a prime). It is shown that these Lie algebras have ``tableau'' representation that completely characterize these algebras. We define for such representation the technical notions analogously to Sylow p-subgroups of symmetric groups of degree $p^n$. Using these notions we characterize the lower and the upper central series and commutant series of these Lie algebras.

1. Холл М. Теория групп.-- М.: ИЛ, 1967.

2. Kaloujnine L. La structure des p-groupes de Sylow des groupes symmetriques finis // Ann. Sci l'Ecole Normal Superior.-- 1948.-- V. 65.-- P. 239-276.

3. Калужнин Л. А. Избранные главы теории групп.-- Киев: Изд-во КГУ, 1979.-- 32~с.

4. Сущанський В. І., Сікора В. С. Операції на групах підстановок. Теорія і застосування.-- Чернівці, ``Рута'', 2003.-- 255~с.

5. Магнус В., Каррас А., Солитер Д. Комбинаторная теория групп.-- М.: Наука, 1974.

6. Huppert B., Blackburn N. Finite groups. V.2.-- Berlin etc.: Springer, 1982.

7. Leedham-Green C. R., Mc Kay S. The structure of groups of Prime Power order.-- London Math. by Monographs, New Series. -- V.27.-- Oxford Science Publication, 2002.-- 334~p.

8. Vaughan M., Lie Methods in group Theory // In: Group St. Andrews 2001 in Oxford, V. II, Cambridge: Camb. Univ. press, 2003.-- P. 547-585.

9. Zelmanov E., Nil Rings and Periodic Groups.-- The Korean Math. Soc. Lecture Notes in Math..-- Seoul, 1992.

10. Gupta C. K., Sushchansky V. I. Lie algebra associated with the group of finitery automorphisms of a p-adic tree // Preprint. Univ. Manitoba, 2000.-- 20~p.

11. Lazard M. Sur le groups nilpotents et les anneaux de Lie // Annal Sci de l'Ecole Norm. Super. (3).-- 1954.-- T.71.-- P. 101-190.

	Lie algebras associated with Sylow $p$-subgroups of finite symmetric groups (in Russian)
Author	V.I.Sushchansky, N.V.Netreba Wital.Suszczanski@polsl.pl, netr@univ.kiev.ua Institute of Mathematics, Silesian University of Technology, Taras Shevchenko National University of Kyiv, Department of Mechanics and Mathematics
Abstract	We study Lie algebras associated with Sylow p-subgroups of symmetric group of degree $p^n$ (p is a prime). It is shown that these Lie algebras have ``tableau'' representation that completely characterize these algebras. We define for such representation the technical notions analogously to Sylow p-subgroups of symmetric groups of degree $p^n$. Using these notions we characterize the lower and the upper central series and commutant series of these Lie algebras.
Keywords	Lie algebra, Sylow p-subgroup, symmetric group
DOI	doi:10.30970/ms.24.2.127-138
Reference	1. Холл М. Теория групп.-- М.: ИЛ, 1967. 2. Kaloujnine L. La structure des p-groupes de Sylow des groupes symmetriques finis // Ann. Sci l'Ecole Normal Superior.-- 1948.-- V. 65.-- P. 239-276. 3. Калужнин Л. А. Избранные главы теории групп.-- Киев: Изд-во КГУ, 1979.-- 32~с. 4. Сущанський В. І., Сікора В. С. Операції на групах підстановок. Теорія і застосування.-- Чернівці, ``Рута'', 2003.-- 255~с. 5. Магнус В., Каррас А., Солитер Д. Комбинаторная теория групп.-- М.: Наука, 1974. 6. Huppert B., Blackburn N. Finite groups. V.2.-- Berlin etc.: Springer, 1982. 7. Leedham-Green C. R., Mc Kay S. The structure of groups of Prime Power order.-- London Math. by Monographs, New Series. -- V.27.-- Oxford Science Publication, 2002.-- 334~p. 8. Vaughan M., Lie Methods in group Theory // In: Group St. Andrews 2001 in Oxford, V. II, Cambridge: Camb. Univ. press, 2003.-- P. 547-585. 9. Zelmanov E., Nil Rings and Periodic Groups.-- The Korean Math. Soc. Lecture Notes in Math..-- Seoul, 1992. 10. Gupta C. K., Sushchansky V. I. Lie algebra associated with the group of finitery automorphisms of a p-adic tree // Preprint. Univ. Manitoba, 2000.-- 20~p. 11. Lazard M. Sur le groups nilpotents et les anneaux de Lie // Annal Sci de l'Ecole Norm. Super. (3).-- 1954.-- T.71.-- P. 101-190.
Pages	127-138
Volume	24
Issue	2
Year	2005
Journal	Matematychni Studii
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