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.