Cyclic by prime fixed point free action

Abstract:

Let the finite group $A$ be acting on a (solvable) group $G$ and suppose that no non-trivial element of $G$ is fixed under the action of all the elements of $A$. Assume furthermore that $(|A| , |G|) = 1$. A long standing conjecture is that then the Fitting height of $G$ is bounded by the length of the longest chain of subgroups of $A$. Even though this conjecture is known to hold for large classes of groups $A$, it is still unknown for some relatively uncomplicated groups. In the present paper we prove the conjecture for all finite groups $A$ that have a normal cyclic subgroup of square free order and prime index. Since many of these groups have natural modules where they act faithfully and coprimely but without regular orbits, the result is new for many of the groups we consider.