Hall higman type theorems

In this work we studied the paper of T.A. Berger entitled "Hall Higman type theorems, I" and Theorem 2.2 in the paper" Hall Higman Type Theorems, Vl" of the same author (Seejll] and [2l ). In the first paper, 1) G = AB is a group with normal cyclic subgroup B and nilpotent complement A where A n B = 1. 2) k = GF(r) is a field for a prime r, and 3) V is a faithful irreducible k [Gj - module such that Vİ is homogeneous for all LAG. L Under these special cases, the conditions for A to have at least three regular orbits on V have been investigated. In Theorem 2.2 in the second paper, A is a nontrivial nilpotent ZQ ^ Z -free group for all primes p| İAİ and k = GP(r) for a prime r||A| and V a faithful k[A] module. Under these conditions, it has been shown that A has at least one regular orbit on V. In this thesis, we have prepared a detailed and readable account of the above mentioned works of Berger, we have clarified many points and filled in many gaps.