Generalized fixed point free automorphisms. (English) Zbl 0605.20015
In a previous paper [J. Algebra 105, 365-371 (1987; Zbl 0604.20021)] the author obtained the best possible bound for the Fitting length of a finite solvable group G with \(G\neq H_{p^ n}(G)\) where \(H_{p^ n}(G)=<x\in G|\) \(x^{p^ n}\neq 1>\) is the generalized Hughes subgroup. In the present article the author modifies his arguments from that paper to obtain a stronger result: Let G be a finite solvable group having a normal subgroup \(K\neq G\) such that every element in \(G\setminus K\) is a p-element, p an odd prime. If \(G\setminus K\) contains an element of order \(p^ n\), then the Fitting length of K is at most \(n+1\) and this bound is best possible. (In the previous paper \(''G\neq H_{p^ n}(G)''\) implies that elements in \(G\setminus H_{p^ n}(G)\) are of order \(p^ n.)\) For \(p=2\) the analogue of the result above is false, as shown by means of an example.
Reviewer: E.Huhro
MSC:
20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
20D25 | Special subgroups (Frattini, Fitting, etc.) |
20D45 | Automorphisms of abstract finite groups |
Citations:
Zbl 0604.20021References:
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