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The principal object of this thesis is to verify if a group that satisfies the normalizer condition does contain a nontrivial abelian normal subgroup. The main reason proposing the topic is that at the time we wrote the report we had no clue as to whether the question had been answered or not.To work on this we proposed the theme （studies on Nilpotent and Soluble groups） which is structured on three chapters.Chapter zero of this thesis gives a brief introduction to the fundamental concepts of group theory-here we collect almost all the information that the rest of this thesis requires.In chapter one we shall study ways in which a group may be decomposed into a set of groups each of which is in some sense of simpler type. We shall develop further the theory introduced in chapter zero of the normal structure of a group. We shall prove the Jordan -Holder theorem and introduce two important classes that are objects of this thesis, the classes of Nilpotent groups and soluble groups. The most important generalizations of commutativity are solubility and Nilpotency. Soluble groups are those that can be constructed from abelain groups by means of a finite number of successive extensions.Not all groups are soluble, for it is clear that nonabelian simple groups are insoluble.A central series is certainly an abelian series and therefore all nilpotent groups are soluble. However, soluble groups are not necessarily nilpotent.For example, let G = S3 and let H be the unique subgroup of G of order 3. Then 1 H G is an abelian series of G , and therefore S3 is soluble. On other hand S3 is not nilpotent, for Z（S3） - 1 and therefore S3 cannot have a central series.Nilpotent groups form a class smaller than that of soluble groups but larger than that of abelian groups. Their definition is more complicated, but they can be more intimately studied than soluble groups. In this chapter we shall prove that a Nilpotent group G satisfies the normalizer condition and we shall use 1. 2. 7 in replying to our principal object.In chapter two we shall talk about generalizations of Nilpotent groups. Of the teeming generalizations of Nilpotence we mention only: local nilpotence and the normalizer condition.