Finite Fixed Point Free Automorphism Groups

Preface A famous theorem by Frobenius in 1901 proves that if a group G contains a proper non trivial subgroup H such that H ∩ g −1 Hg = {1 G } for all g ∈ G \ H, then there exists a normal subgroup N such that G is the semidirect product of N and H. Groups with this property-the so called Frobenius groups-arise in a natural way as transitive permutation groups, but they can also be characterized as semidirect product of a group N and a group of automorphisms H acting on N with the identity of N as single fixed point. Only a short time later in 1905, Dickson obtained the first proper nearfields, when he " distorted " the multiplication in a finite field. In 1936, Zassenhaus made advance of the fact that for all elements a in a right nearfield N the mappings λ a : x → ax with nearfield multiplication are automorphisms without non trivial fixed points on (N, +) and determined the structure of all finite fixed point free automorphism groups. This enabled him to characterize all finite nearfields as Dickson nearfields up to 7 exceptional cases. Whereas the additive group of a finite nearfield is elementary abelian, Thompson managed to show that any group which admits a fixed point free automorphism of prime order has to be nilpotent in 1959. Planar nearrings (N, +, ·) can be regarded as generalized nearfields and Fer-rero's discovery that every planar nearring can be constructed from an additive group (N, +) and a fixed point free automorphism group acting thereupon does not come as a surprise at all. In the finite case Frobenius groups, nearfields and planar nearrings can be interpreted as different aspects of the same group theoretical concept: a nilpotent group with a fixed point free automorphism group. We present these interrelations in Chapter 7. Although a lot is known about the structure of fixed point free automorphism groups in theory, the determination of such a group Φ for an arbitrary nilpotent group G is not trivial at all. The objective of this thesis is to give a theoretical setting how to construct fixed point free automorphism groups and to provide functions for actually doing this by computer. The main tool used in this computational context is extension. After revisiting the classical results from Thompson and Zassenhaus in Chapter 3, where we also …