Here is a logic-based viewpoint on the use of isomorphisms and homomorphisms. Every first-order structure (e.g. group, ring, field, module, ...) has an associated (complete) theory, namely the set of all sentences in its language that are true for it. For example, each group satisfies the group a...

The existing answers are about groups, but here is a more general perspective. Suppose we have some kind of structure $S$ with a substructure $T$, meaning that $T$ inherits the operations and predicates from $S$ and is closed under those operations. We could ask the general question of whether...