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A nilpotent ideal in the Lie rings with automorphism of prime order
N. Yu. Makarenko Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
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.
Keywords:
Lie rings, Lie algebras, automorphisms of Lie rings, automorphisms of Lie algebras, almost regular automorphisms, graded Lie rings, graded Lie algebras.
UDC:
512.5
Received: 07.06.2005