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Centralizers of finite nilpotent subgroups in locally finite groups. (English. Russian original) Zbl 0870.20025
Algebra Logic 35, No. 4, 217-228 (1996); translation from Algebra Logika 35, No. 4, 389-410 (1996).
The authors study the structure of locally finite groups containing a finite subgroup whose centralizer is similar to a linear group. A locally finite group $$G$$ is said to be a $$\lambda$$-group if $$G$$ contains, as a subgroup of finite index, the central product of a Chernikov group and a finite number of quasisimple groups of Lie type.
The main result is as follows: Theorem 0.2. Let $$G$$ be a locally finite group containing a finite nilpotent group whose centralizer is a $$\lambda$$-group. Suppose that the Hirsch-Plotkin radical of $$G$$ (i.e. maximal normal locally nilpotent subgroup) is trivial. Then $$G$$ contains a subgroup of finite index whose commutator subgroup is isomorphic to the direct product of finitely many simple groups of Lie type.
The proof uses the classification of the finite simple groups and is based on the fact (Theorem 0.11) that a finite group admitting a nilpotent regular automorphism group is solvable.

##### MSC:
 20F50 Periodic groups; locally finite groups 20G15 Linear algebraic groups over arbitrary fields 20E07 Subgroup theorems; subgroup growth 20E25 Local properties of groups 20D06 Simple groups: alternating groups and groups of Lie type
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