A group homomorphism is a function between two groups that preserves the group operation. This means that if you take two elements from the first group, apply the group operation, and then map the result to the second group, it will be the same as mapping each element first and then applying the operation in the second group. Formally, if f: G \to H is a homomorphism, then for any elements a, b in group G , it holds that f(ab) = f(a)f(b) . Homomorphisms are important in the study of algebraic structures because they allow mathematicians to understand how different groups relate to each other. They can help identify properties that are preserved under the mapping, such as the identity element and inverses. Examples of groups include integers under addition and non-zero rational numbers under multiplication, both of which can have homomorphisms defined between them.