The finite simple groups started attracting the interest of the mathematicians in the nineteenth century, especially once the concept of normal subgroups was introduced by Galois in 1832; di erentiation between the simple and compound groups by Camille Jordan in 1870; and the theorems on subgroups of prime power order published by Ludwig Sylow in 1872. This was given in a historical form as a means of introduction. This thesis also focuses on the Sylow's theorem and their wide range of use in classifying nite groups in algebra. Groups of order 1-15 were classi ed using the Sylow's theorems in addition to other established results in algebra. The uniqueness and existence of such groups were also proved to the best of the writer's ability.
