Abstract
We improve the conclusion in Khukhro's theorem stating that a Lie ring (algebra) L admitting an automorphism of prime order p with finitely many m fixed points (with finite-dimensional fixed-point subalgebra of dimension m) has a subring (subalgebra) H of nilpotency class bounded by a function of p such that the index of the additive subgroup |L: H| (the codimension of H) is bounded by a function of m and p. We prove that there exists an ideal, rather than merely a subring (subalgebra), of nilpotency class bounded in terms of p and of index (codimension) bounded in terms of m and p. The proof is based on the method of generalized, or graded, centralizers which was originally suggested in [E. I. Khukhro, Math. USSR Sbornik 71 (1992) 51–63]. An important precursor is a joint theorem of the author and E. I. Khukhro on almost solubility of Lie rings (algebras) with almost regular automorphisms of finite order.
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Original Russian Text Copyright © 2005 Makarenko N. Yu.
The author was supported by the Program “Universities of Russia” (Grant UR.04.01.202).
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Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 1360–1373, November–December, 2005.
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Makarenko, N.Y. A Nilpotent Ideal in the Lie Rings with Automorphism of Prime Order. Sib Math J 46, 1097–1107 (2005). https://doi.org/10.1007/s11202-005-0104-0
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DOI: https://doi.org/10.1007/s11202-005-0104-0