In this chapter, we shall consider a process of constructing a new group from a finite number of given groups. Actually, the process (the external direct product) can be extended to the case where an infinite number of groups are given, but we shall not go into these matters here. At the same time, we shall consider the intimately related situation of decomposing a given group in a certain fashion (the internal direct product) into a product (with the usual meaning) of a finite number of subgroups. We shall investigate the relationship between this situation (internal direct product) and the first (external direct product) of our considerations. We shall see that up to an isomorphism the two concepts of direct product are indistinguishable. In the second section, we shall consider applications of this construction.