In the first section, we shall consider special subgroups of a group called normal subgroups which are quite important in all of group theory. We shall see that normal subgroups enable us to construct new groups using the set of cosets relative to these subgroups. Finally, the last section in this chapter considers a class of groups which contains no proper normal subgroups. Such groups are called simple groups. In recent years the problem of determining (or classifying) all finite simple groups has received more attention from group theorists than any other single problem. (See the article by D. Gorenstein [G].) The simple groups are important because they play a role in finite group theory somewhat analogous to that of the primes in number theory. As a matter of fact, the classification of finite simple groups has been completed. Its proof involves some 500 journal articles covering approximately 15,000 printed pages. In the words of D. Gorenstein this ``... is unprecedented in the history of mathematics ...''