Fixed Point Free Action with Some Regular Orbits

AbstractLet the finite groupAbe acting on a finite (solvable) groupGand suppose that no non-trivial element ofGis fixed under the action of all the elements ofA. Assume furthermore that (|A|,|G|)=1. A long-standing conjecture is that then the Fitting height ofGis bounded by the length of the longest chain of subgroups ofA. In3, this was proved in the case where for every proper subgroupBofAand everyB-invariant elementary abelian sectionSofG, there exists somev∈Ssuch thatCB(v)=CB(S) (we say thatBhas a regular orbit onS). In the present paper we establish the conjecture assuming only that some of these sections have regular orbits