A Constructive Method for Using Known Groups as Building Blocks to form More Groups

In this paper, we present some methods of constructing new groups from given ones. We first examined the direct product of the finite groups Gi for n³2 and then observed that . We also established the fact that for any prime number p > 2 and any positive integer k, |U(pk)| = |U(2pk)| where U(n) is the set of all integers less than n and relatively prime to n, and finally conclude on symmetries by constructing groups and their respective subgroups, characteristics and the unique factorization of the elements.