Abstract
A Liep-algebraL is calledn-power closed if, in every section ofL, any sum ofp i+nth powers is ap ith power (i>0). It is easy to see that ifL isp n-Engel then it isn-power closed. We establish a partial converse to this statement: ifL is residually nilpotent andn-power closed for somen≥0 thenL is (3p n+2+1)-Engel ifp>2 and (3 · 2n+3+1)-Engel ifp=2. In particular, thenL is locally nilpotent by a theorem of Zel’manov. We deduce that a finitely generated pro-p group is a Lie group over thep-adic field if and only if its associated Liep-algebra isn-power closed for somen. We also deduce that any associative algebraR generated by nilpotent elements satisfies an identity of the form (x+y)p n=x p n+y p n for somen≥1 if and only ifR satisfies the Engel condition.
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References
Yu. Bahturin,Identical Relations in Lie Algebra, VNU Science Press, 1987.
A. Braun,The nilpotency of the radical in finitely generated PI-rings, Journal of Algebra89 (1984), 375–396.
J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal,Analytic Pro-p Groups, London Mathematical Society Lecture Note Series157, Cambridge University Press, 1991.
M. Lazard,Groupes analytiques p-adiques, Publications Mathématiques de l’Institute Hautes Études Scientifiques26 (1965).
M. Lincoln and D. Towers,Frattini theory for restricted Lie algebras, Archiv der Mathematik45 (1985), 101–110.
A. Lubotzky and A. Mann,Powerful p-groups I: finite groups, Journal of Algebra105 (1987), 484–505.
A. Lubotzky and A. Mann,Powerful p-groups II: p-adic analytic groups, Journal of Algebra105 (1987), 506–515.
A. Mann,On the power structure of p-groups, I, Journal of Algebra42 (1976), 121–135.
D. M. Riley,Analytic pro-p groups and their graded group rings, Journal of Pure and Applied Algebra90 (1993), 69–76.
D. M. Riley,Algebras generated by nilpotent elements of bounded index, preprint, 1995.
D. M. Riley and J. F. Semple,Completion of restricted Lie algebras, Israel Journal of Mathematics86 (1994), 277–299.
A. Shalev,On associative algebras satisfying the Engel condition, Israel Journal of Mathematics67 (1989), 287–290.
A. Shalev,Characterization of p-adic analytic groups in terms of wreath products, Journal of Algebra145 (1992), 204–208.
A. Shalev,Polynomial identities in graded group rings, restricted Lie algebras, and p-adic analytic groups, Transactions of the American Mathematical Society337 (1993), 451–462.
H. Strade and R. Farnsteiner,Modular Lie Algebras, Marcel Dekker, New York, 1988.
E. I. Zel’manov,Solution of the restricted Burnside problem for groups of odd exponent, Mathematics of the USSR-Izvestiya36 (1991), 41–60.
E. I. Zel’manov,Solution of the restricted Burnside problem for 2-groups, Mathematics of the USSR-Sbornik72 (1992), 543–565.
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This project was supported by the CNR in Italy and NSF-EPSCoR in Alabama during the first author’s stay at the Università di Palermo.
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Riley, D.M., Semple, J.F. Power closure and the Engel condition. Isr. J. Math. 97, 281–291 (1997). https://doi.org/10.1007/BF02774041
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DOI: https://doi.org/10.1007/BF02774041