Fitting height of a finite group with a Frobenius group of automorphisms.
Let \(G\) be a finite group admitting a Frobenius group of automorphisms \(FH\) (here \(F\) is the kernel, \(H\) is the complement) such that \(C_G(F)=1\). A theorem of \textit{V. V. Belyaev} and \textit{B. Hartley} [Algebra Logic 35, No. 4, 217-228 (1996); translation from Algebra Logika 35, No. 4, 389-410 (1996; Zbl 0870.20025)] ensures that \(G\) is solvable. The author of the paper under review is interested in deriving information on the structure of \(G\) given information on the structure of \(C_G(H)\), which is the subgroup of \(G\) fixed by the complement \(H\). The main Theorem 2.1 states that: a) \(F_i(C_G(H))=F_i(G)\cap C_G(H)\) for every term \(F_i\) of the Fitting series of \(G\), and b) the Fitting heights of \(G\) and \(C_H(G)\) are equal. A similar result holds for the \(\pi\)-lengths of \(G\) and \(C_G(H)\). -- The bulk of the proof is to establish the nontrivial inclusion in part a) of Theorem 2.1 for the most important case \(i=1\). For this, the author uses a large number of results on \(FH\)-modules which have independent interest.
scientific article; zbMATH DE number 6140809
scientific article