Study of Isomorphism, Homomorphism and their application

Abstract

In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished. In this paper we present the first stages of constructing a systematic method for classifying groups of small orders. Classifying groups usually arise when trying to distinguish the number of non-isomorphic groups of order n. This paper arose from an attempt to find a formula or an algorithm for classifying groups given invariants that can be readily determined without any other known assumptions about the group. This formula is very useful if we want to know if two groups are isomorphic. Mathematical objects are considered to be essentially the same, from the point of view of their algebraic properties, when they are isomorphic. A homomorphism is a map between two groups which respects the group structure. More formally, let G and H be two group, and f a map from G to H (for every g∈G, f(g)∈H). Then f is a homomorphism if for every g1,g2∈G, f(g1g2)=f(g1)f(g2). if H<G, then the inclusion map i(h)=h∈G is a homomorphism. Another example is a homomorphism from Z to Z given by multiplication by 2, f(n)=2n. .