An automorphism group V
acting on a group G is said to be without fixed points if for any g ∈ G, v(g) = g for
all v ∈ V implies that g = 1. The structure of V in this case has been shown to
influence the structure of G. For example if V is cyclic of order p and G finite
then John Thompson has shown that G must be nilpotent. Gorenstein and
Herstein have shown that if V is cyclic of order 4 then a finite group G must be
solvable of p-length 1 for all p||G| and G must possess a nilpotent commutator
subgroup.
In this paper we will consider the case where G is finite and V noncyclic of order
4. Since V is a two group all the orbits of G under V save the identity have order a
positive power of 2. Thus G is of odd order and by the work of Feit-Thompson G is
solvable. We will show that G has p-lengh 1 for all p||G| and G must possess a
nilpotent commutator subgroup.
