A group homomorphism is a mathematical function between two groups that preserves the group operation. This means 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 \rightarrow H is a homomorphism, then for any elements a, b in group G , it holds that f(a \cdot b) = f(a) \cdot f(b) . Homomorphisms are important in the study of algebraic structures because they help us understand how different groups relate to each other. For example, if G and H are groups, and f is a homomorphism, then the image of f is a subgroup of H . This concept is widely used in