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On fixed points of automorphisms of Lie rings and locally finite groups. (English. Russian original) Zbl 0901.20026
Algebra Logic 34, No. 6, 395-405 (1995); translation from Algebra Logika 34, No. 6, 706-723 (1995).
The authors investigate the structure of locally finite groups with conditions on fixed points of automorphisms. They obtain the following important result: if a locally finite group \(P\) of prime period has a finite solvable group of automorphisms \(G\) such that the corresponding fixed point subgroup \(P^G\) is solvable, then \(P\) is nilpotent. Its index of nilpotency is bounded above by a function which depends on the period of \(P\), the order of \(G\) and the index of solvability of \(P^G\). This function can be explicitly approximated. Both conditions, the local finiteness and the solvability of the fixed point subgroup, are necessary. The proof has two main ingredients: a generalization of V. A. Kreknin’s result [Dokl. Akad. Nauk SSSR 150, 467-469 (1963; Zbl 0134.03604)] on solvability of Lie rings having a regular automorphism of finite order, and a generalization of P. J. Higgins’ result [Math. Proc. Camb. Philos. Soc. 50, 8-15 (1954; Zbl 0055.02601)] on nilpotency of solvable Lie algebras with Engel condition.

MSC:
20F50 Periodic groups; locally finite groups
20F28 Automorphism groups of groups
20F40 Associated Lie structures for groups
20F45 Engel conditions
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