Finite P-groups With Small |Kern(G)|

It is important in the study of finite groups that studying the influence of the number of irreducible characters on the structure of a finite group G.Many scholars studied finite groups in which the number of non-linear irreducible characters is at most 2.It should be pointed out that if |Kern(G)|?3,then the number of non-linear irreducible characters of G is boundless.In term of |Kern(G)|,the finite groups scholars studied,up to now,are of |Kern(G)|?3 and very specific.Hence it is more general and interesting to study the structure of finite groups in term of |Kern(G)|.In 1997,Berkovich and Zhmud proposed the following problem in their book "Characters of finite groups Part 2":Question 2.Classify the groups G such that |Kern(G)|?3 or |QKern(G)|?3.The structure of finite groups with |Kern(G)|?1 was determined by Berkovich and Zhmud.However,it is an unsolvable problem to determine the structure of finite groups with |Kern(G)|=2 and |Kern(G)|=3 respectively.In this thesis,we study the problem and determine the structure of finite p-groups with |Kern(G)|=2 and |Kern(G)|=3 respectively.Hence the problem is solved partly.