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## 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 for all , 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 .

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:
1. All Sylow groups of  are cyclic or generalized quaternion groups.

2. If has even order, it contains a unique element  of order two. The kernel  is abelian and for all .
3. If has odd order, it is metacyclic, i.e., it has a cyclic normal subgroup with cyclic factor group. If 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 . The subgroup of GL that is generated by the matrices and operates fixed-point-freely on by multiplication. 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.

Next: 2.1.4 Positioning of this Up: 2.1 State of the Previous: 2.1.2 Introduction to near-rings
Peter Mayr 2002-08-12