Frobenius groups of automorphisms with almost fixed point free kernel

Abstract

Let $FH$ be a Frobenius group with kernel $F$ and complement $H$, acting coprimely on the finite solvable group $G$ by automorphisms. We prove that if $C_G(H)$ is of Fitting length $n$ then the index of the $n$-th Fitting subgroup $F_n(G)$ in $G$ is bounded in terms of $|C_G(F)|$ and $|F|.$ This generalizes a result of Khukhro and Makarenko \cite{k-m} which handles the case $n=1.$