Almost solubility of Lie algebras with almost regular automorphisms

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The well-known theorem of Borel–Mostow–Kreknin on solubility of Lie algebras with regular automorphisms is generalized to the case of almost regular automorphisms. It is proved that if a Lie algebra L admits an automorphism ϕ of finite order n with finite-dimensional fixed-point subalgebra of dimension dim C L ( ϕ)= m, then L has a soluble ideal of derived length bounded by a function of n whose codimension is bounded by a function of m and n (Theorem 1). A virtually equivalent formulation is in terms of a ( Z/n Z) -graded Lie algebra L whose zero component L 0 has finite dimension m. The functions of n and of m and n in Theorem 1 can be given explicit upper estimates. The proof is of combinatorial nature and uses the criterion for solubility of Lie rings with an automorphism obtained in [E.I. Khukhro, Siberian Math. J. 42 (2001) 996–1000]. The method of generalized, or graded, centralizers is developed, which was originally created in [E.I. Khukhro, Math. USSR Sbornik 71 (1992) 51–63] for almost regular automorphisms of prime order. As a corollary we prove a result analogous to Theorem 1 on locally nilpotent torsion-free groups admitting an automorphism of finite order with the fixed points subgroup of finite rank (Theorem 3). We also prove an analogous result for Lie rings with an automorphism of finite order having finitely many fixed points (Theorem 2).