Generalized fixed point free automorphisms (Q1084177)

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 \underbar{all} 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.