Lie algebras associated with wreath products of elementary abelian groups

N.V.Bondarenko

We study Lie algebras associated with the lower central series of the wreath product $P_{m,n}$ of $m$ copies of elementary abelian $p$-groups of degree $n$. It is shown that these Lie algebras have special "tableau" representation. We define a wreath product of a Lie algebra $L$ with an abelian finite-dimensional Lie algebra over the field $\mathbb{F}_p$. We prove that the Lie algebra associated with the lower central series of the group $P_{m,n}$ is isomorphic to the wreath product of $m$ copies of the abelian Lie algebra of dimension $n$ over the field $\mathbb{F}_p$.

1. Bahturin Y.A. Identical Relations in Lie Algebras, Nauka, Moskow, 1985; \ VNU Scientific Press, Utrecht, 1987.

2. Bartoldi L., Grigorchuk R.I. Lie Methods in Growth of Groups and Groups of Finite Width, Computational and Geometric Aspects of Modern Algebra. Cambridge: Camb. Univ. Press, 2000, p. 1--28.

3. Bartoldi L. Lie Algebras and Grouth in Branch Groups, Pasific Journal of Mathematics, Vol. 218, No. 2, 2005, p. 241--282.

4. Huppert B., Blackburn N., Finite groups, V.2, Berlin etc., Springer, 1982.

5. Leedham-Green C.R., McKay S., The Structure of Groups of Prime Power Order, London Math. by Monographs, New Series, 27 Oxford Science Publication, 2002, 334~p.

6. Kaloujnine L. La structure des $p$--groupes de Sylow des groupes symetriques finis, Ann. Sci de l'Ecole Normal Superiore, Vol. 65, N. 3, 1948, p.~239--276.

7. Kaloujnine L.A., Sushchansky V.I. Wreath products of abelian groups, \ Trans. Mosc. Math. Soc., \ V. 29, 1976, p.~145--160.

8. Lazard M. Sur les groupes nilpotents et les anneaux de Lie, Annal Sci de l'Ecole Norm. Super. (3) 71, 1954, p.~101--190.

9. Magnus W., \"Uber Gruppen und zugeordnete Liesche Ringer, J. Reine Angew. Math., 182, 1940, p.~142--149.

10. Magnus W., Karras A., Soliter D. Combinatorial group theory, Interscience Publishers, New York-London-Sydney, 1966, 444~p.

11. Netreba N.V., Sushchansky V.I. Lie algebras associated with Sylow p-subgroups of finite symmetric groups, 5th International Algebraic Conference in Ukraine, Abstracts, Odessa, 2005, p.~142.

12. Shmelkin A.L., Wreath Product of Lie Algebras and their Application to Group Theory, Trudy Moskovs. Matem. O-va., V. 29, 1973, p.~247-260 (in Russian).

13. Sushchansky V.I., Wreath products of elementary abelian groups, Trans. Math. Notes, V. 11, 1972, p.~41--47.

14. Sushchansky V.I. The lower central series of Lie ring of Sylow p-subgroup of $I_sZ_p$, IV All-Union school of Lie algebra and their applications in mathematics and physics, Abstract, Kazan, 1990, p.~44 (in Russian).

15. Sushchansky V.I., Netreba N.V., Wreath product of Lie algebras and Lie algebras associated with Sylow p-subgroups of finite symmetric groups, Algebra and Discrete Mathematics, N. 1, 2005, p.~ 122--132.

16. Vaughan M., Lie Methods in group Theory, In: Group St. Andrews 2001 in Oxford, v. II, Cambridge: Camb. Univ. press, 2003, p.~547--585.

17. Zelmanov E. Nil Rings and Periodic Groups, The Korean Mathematical Society, 1992, 72~p.

	Lie algebras associated with wreath products of elementary abelian groups
Author	N.V.Bondarenko netr@univ.kiev.ua Kyiv Taras Shevchenko National University,Faculty of mechanics and mathematics
Abstract	We study Lie algebras associated with the lower central series of the wreath product $P_{m,n}$ of $m$ copies of elementary abelian $p$-groups of degree $n$. It is shown that these Lie algebras have special "tableau" representation. We define a wreath product of a Lie algebra $L$ with an abelian finite-dimensional Lie algebra over the field $\mathbb{F}_p$. We prove that the Lie algebra associated with the lower central series of the group $P_{m,n}$ is isomorphic to the wreath product of $m$ copies of the abelian Lie algebra of dimension $n$ over the field $\mathbb{F}_p$.
Keywords	Lie algebra, wreath product, lower central series
DOI	doi:10.30970/ms.26.1.3-16
Reference	1. Bahturin Y.A. Identical Relations in Lie Algebras, Nauka, Moskow, 1985; \ VNU Scientific Press, Utrecht, 1987. 2. Bartoldi L., Grigorchuk R.I. Lie Methods in Growth of Groups and Groups of Finite Width, Computational and Geometric Aspects of Modern Algebra. Cambridge: Camb. Univ. Press, 2000, p. 1--28. 3. Bartoldi L. Lie Algebras and Grouth in Branch Groups, Pasific Journal of Mathematics, Vol. 218, No. 2, 2005, p. 241--282. 4. Huppert B., Blackburn N., Finite groups, V.2, Berlin etc., Springer, 1982. 5. Leedham-Green C.R., McKay S., The Structure of Groups of Prime Power Order, London Math. by Monographs, New Series, 27 Oxford Science Publication, 2002, 334~p. 6. Kaloujnine L. La structure des $p$--groupes de Sylow des groupes symetriques finis, Ann. Sci de l'Ecole Normal Superiore, Vol. 65, N. 3, 1948, p.~239--276. 7. Kaloujnine L.A., Sushchansky V.I. Wreath products of abelian groups, \ Trans. Mosc. Math. Soc., \ V. 29, 1976, p.~145--160. 8. Lazard M. Sur les groupes nilpotents et les anneaux de Lie, Annal Sci de l'Ecole Norm. Super. (3) 71, 1954, p.~101--190. 9. Magnus W., \"Uber Gruppen und zugeordnete Liesche Ringer, J. Reine Angew. Math., 182, 1940, p.~142--149. 10. Magnus W., Karras A., Soliter D. Combinatorial group theory, Interscience Publishers, New York-London-Sydney, 1966, 444~p. 11. Netreba N.V., Sushchansky V.I. Lie algebras associated with Sylow p-subgroups of finite symmetric groups, 5th International Algebraic Conference in Ukraine, Abstracts, Odessa, 2005, p.~142. 12. Shmelkin A.L., Wreath Product of Lie Algebras and their Application to Group Theory, Trudy Moskovs. Matem. O-va., V. 29, 1973, p.~247-260 (in Russian). 13. Sushchansky V.I., Wreath products of elementary abelian groups, Trans. Math. Notes, V. 11, 1972, p.~41--47. 14. Sushchansky V.I. The lower central series of Lie ring of Sylow p-subgroup of $I_sZ_p$, IV All-Union school of Lie algebra and their applications in mathematics and physics, Abstract, Kazan, 1990, p.~44 (in Russian). 15. Sushchansky V.I., Netreba N.V., Wreath product of Lie algebras and Lie algebras associated with Sylow p-subgroups of finite symmetric groups, Algebra and Discrete Mathematics, N. 1, 2005, p.~ 122--132. 16. Vaughan M., Lie Methods in group Theory, In: Group St. Andrews 2001 in Oxford, v. II, Cambridge: Camb. Univ. press, 2003, p.~547--585. 17. Zelmanov E. Nil Rings and Periodic Groups, The Korean Mathematical Society, 1992, 72~p.
Pages	3-16
Volume	26
Issue	1
Year	2006
Journal	Matematychni Studii
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