A Derived Method for Construction and Classification of Morphisms and Representations Between Finite Groups

This paper aimed at constructing new group presentations from known presentations using direct product of two or more groups. We established the fact that for any prime number p > 2 and any positive integer n, |U(p n )| = |U(2p n )| and then used symmetries to construct groups and their respective subgroups, characteristics and the unique factorization of the elements. Functions fi on finite group G such that each fi is a morphism are constructed and the fact that if G is any finite Abelian group, H a subgroup of G, then the factor group G/H is a finite Abelian group is proved. We finally established that if |G| = n such that n = r,s,t, then G  Zr  Zs  Zt where r,s,tZ + and then identify some homomorphism and automorphism on finite groups by listing all the possible maps from the group to itself with the help of GAP.