Fixed Point Free Action with Some Regular Orbits Original Research Article

Let 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. In[3], this was proved in the case where for every proper subgroupBofAand everyB-invariant elementary abelian sectionSofG, there exists somev set membership, variant 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.