The Finite Groups With A Prime Number And Same Degree Of Nonlinear Irreducible Characters

The representation of finite groups is an important, branch of mordern mathematics. Character theory is one of the main tools of the representation of finite groups over the complex field, it has close relationship with the structure of finite groups. In fact, how the number and the degree of nonlinear irreducible characters of finite groups influnce the structure of groups is an important issue of category theory. This academic dissertation focuses on these problems.In this academic dissertation, we study the groups which have a prime number and same degree of nonlinear irreducible characters over the complex field, in these conditions, we classify the finite groups when they are nilpotent but not 2-group, also we classify the groups when they are non-nilpotent and the commutator subgroup is minimal. The whole dissertation includes three chapters.In the first chapter, we introduce the groundwork of this text, also the research direction and the trends of the development.In the second chapter, we list some fundamental concepts and facts, including special p-group. extraspecial p-group, semi-extraspecial p-group and Frobenius group, also some important lemmas such as the famous Clifford's correspondence theorem. They give a necessary preparation for the third chapter.In the third chapter, we solve the problems which this academic dissertation focuses on by means of the properties of nilpotent and non-nilpotent groups.The Chapter 3 is divided in two sections: In the first section,we prove that the nilpotent group but not 2-group satisfying our conditions is a direct product of an extraspecial-2 group with a prime oreder group.In the second section, first we prove that the commutator subgroup of a non-nilpotent group which has three nonlinear irreducible characters and their degrees are same is minimal normal. Then we consider more general case. We prove that the groups which satisfying our conditions and whose commutator subgroup is minimal normal are divided into three distinct classes. The first are Frobenius Groups. The second are direct products of a Frobenius group with an elementary abelian group. The third have their factor group G/Z(G) as a Frobenius group.