Cyclic by prime fixed point free action
Alexandre Turull
Proc. Amer. Math. Soc. 125 (1997), 3465-3470.

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.