2.1.3 Introduction to Frobenius groups

In 1901, Frobenius proved the following theorem: If a group

contains a proper non trivial subgroup

such that $H\cap g^{-1}Hg=\{1_{G}\}$ for all $g\in G\setminus H$ , then there exists a normal subgroup

such that

is the semi-direct product of

and

. 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 semi-direct product of a group

and a fixed-point-free group $H\leq \textnormal{Aut}\, N$ .

In 1936, Zassenhaus determined the structure of all finite fixed-point-free automorphism groups. In 1959, Thompson managed to show that any group which admits a fixed-point-free automorphism of prime order has to be nilpotent.

The main structural result on Frobenius complements can be summarized as follows (see [Hup67] V 8.15 and 8.18):

Theorem 1 Let

be a Frobenius group with kernel

and complement

. Then the following hold:

All Sylow groups of are cyclic or generalized quaternion groups.
If has even order, it contains a unique element of order two. The kernel is abelian and $n^h=n^{-1}$ for all $n\in N$ .
If has odd order, it is metacyclic, i.e., it has a cyclic normal subgroup with cyclic factor group. If $h\in H$ has prime order, then the subgroup generated by is normal and either contained in the center or in the derived subgroup .

Let us give one example of a fixed-point-free automorphism group acting on $N = \mathbb{Z}_{13}\times\mathbb{Z}_{13}$ . The subgroup $\Phi$ of GL

that is generated by the matrices $\left( \begin{smallmatrix} 3 & 4 \\ 4 & -3 \end{smallmatrix}\right)$ and $\left( \begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right)$ operates fixed-point-freely on

by multiplication. $\Phi$ is isomorphic to the quaternion group with

elements.

The main point for our purposes is the fact: ``Every planar near-ring determines a Frobenius group, and conversely''. Both theories benefit from each other.