Nilpotent length of a finite group admitting a Frobenius group of automorphisms with fixed-point-free kernel. (English. Russian original) Zbl 1223.20016
Algebra Logic 49, No. 6, 551-560 (2011); translation from Algebra Logika 49, No. 6, 819-833 (2010).
Suppose that a finite group \(G\) admits a Frobenius group \(FH\) of automorphisms with kernel \(F\) and complement \(H\) such that the fixed-point subgroup of \(F\) is trivial, i.e. \(C_G(F)=1\), and the orders of \(G\) and \(H\) are coprime. The author proves that the nilpotent length of \(G\) is equal to the nilpotent length of \(C_G(H)\) and the Fitting series of the fixed-point subgroup \(C_G(H)\) coincides with the series obtained by taking the intersection of \(C_G(H)\) with the Fitting series of \(G\). The author asks whether the condition that the orders of \(G\) and \(H\) are relatively prime is really necessary.
Reviewer: Alexandre Turull (Gainesville)
MSC:
| 20D45 | Automorphisms of abstract finite groups |
| 20D30 | Series and lattices of subgroups |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
Keywords:
Frobenius groups; automorphisms; centralizers; finite groups; soluble groups; nilpotent lengths; Fitting seriesReferences:
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