TITLE: What is the difference between homomorphism and isomorphism?
QUESTION [40 upvotes]: Let $G$ and $H$ be two groups, and $f$ a map from $G$ to $H$ ($\forall g\in G \Rightarrow f(g)\in H$). Then $f$ is a homomorphism if $\forall g_1,g_2\in G \Rightarrow f(g_1g_2)=f(g_1)f(g_2)$.
This means that $G$ and $H$ are algebraically identical.
Isomorphism is a bijective homomorphism.
I see that isomorphism is more than homomorphism, but I don't really understand its power. When we hear about bijection, the first thing that comes to mind is topological homeomorphism, but here we are talking about algebraic structures, and topological spaces are not algebraic structures.
Then what is the "power" that makes us to define isomorphism as a special case of homomorphism?

REPLY [2 votes]: It's true that isomorphism get along extremely well with the specific two objects they are relating, making them for most isomorphism, "identical up to naming" (same up to isomorphism). An important point is that what makes a isomorphism in each area of math is designed specifically in mind to preserve that properties. In general isomorphism need the "iso" part wich means bijective function, to be able to correspond exactly each element of one with one of the other. They also preserve other crucial information, in the case of group theory for exaample, they preserve the product of two elements, wich happens to be everything one needs to know about a group to make any statement about it!