On Frobenius Groups

Frobenius groups are a fundamental class of finite groups, distinguished by their remarkable structural properties and their role in the development of modern group theory. Their study originates from Frobenius’ Theorem, which states that if a finite group G contains a nontrivial proper subgroup H such that H and each of its conjugates outside H intersect trivially, then there exists a normal subgroup N of G such that every element of G can be uniquely expressed as a product of an element of N and an element of H, with N and H having only the identity element in common. In this case, G is called a Frobenius group, with Frobenius complement H and Frobenius kernel N . A central role in the proof of Frobenius’ Theorem is played by character theory and the result represents one of the most remarkable applications of characters to the study of finite groups. The concept of a Frobenius group naturally leads to the notion of a Frobe- nius partition and to the semidirect product decomposition G = N ⋊H, where the complement H acts on the kernel N by fixed-point-free automorphisms. A further perspective on Frobenius groups is given by permutations: a Frobenius action of a group G on a non-singleton set is a transitive but non- regular action in which the stabilizers of any two distinct points intersect trivially. The structural significance of Frobenius groups is further emphasized by Thompson’s theorem, which establishes the nilpotency of the Frobenius ker- nel. The proof of this deep result relies on the theory of p-complements and on the so-called Thompson subgroup. Thompson’s theorem yields that Frobe- nius groups have a remarkably rigid structure: in particular, the Frobenius kernel is uniquely determined, and consequently so is the Frobenius partition. By presenting and analyzing several classical results and examples, this thesis offers a thorough overview of the theory of Frobenius groups and their main structural properties.