Groups of Orders

In this paper unless otherwise stated the letter p represents a fixed prime number. The concept of p- groups is fundamental in the theory of groups. Sylow theorems will be assumed known in this paper. In classifying finite groups we know in the Abelian case that the number of groups of order n is influenced largely by the character of the prime factorization of n, and not by the size of n alone. Any finite group G contains the so called Sylow p-subgroups which are p-groups and are closely linked to the structure of G. Recent developments in theory of finite simple groups have brought insights on p-groups and have suggested investigations in diverse areas. In this paper, however, we shall present some of the most basic results on transitive p-groups and their defining relations