On fixed-point-free automorphisms

Let R be a cyclic group of prime order which acts on the extraspecial group F in such a way that F=[F,R]. Suppose RF acts on a group G so that CG(F)=1 and (|R|,|G|)=1. It is proved that F(CG(R))⊆F(G). As corollaries to this, it is shown that the Fitting series of CG(R) coincides with the intersections of CG(R) with the Fitting series of G, and that when |R| is not a Fermat prime, the Fitting heights of CG(R) and G are equal.