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.
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.
Reviewer: V.D.Mazurov (Novosibirsk)
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 |
Keywords:
\(\lambda\)-groups; locally finite groups; subgroups of finite index; central products; Chernikov groups; quasisimple groups of Lie type; centralizers; Hirsch-Plotkin radicalReferences:
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