How does a homomorphism preserve the operation of an algebraic system?

In modern algebra, a homomorphism preserves the operation of an algebraic system by ensuring that the result of an operation performed on elements in one system corresponds directly to the result of the operation performed on their images in the second system.1

Specifically, if $G$ and $H$ are algebraic systems with operations $\oplus$ and $\otimes$ respectively, a homomorphism is a function $f$ from $G$ to $H$ such that for any elements $g$ and $g^{\prime }$ in $G$:

$$f(g\oplus g^{\prime })=f(g)\otimes f(g^{\prime })$$

This means the "image" of a product is the product of the images. Because this mapping maintains the structural relationships between elements—such as mapping the identity element of one system to the identity of the other and preserving inverses—homomorphisms are referred to as structure-preserving maps. By establishing this correspondence, mathematicians can apply known features of one system to another, which is essential for classifying and analyzing different algebraic structures like groups, rings, or fields.1