TITLE: What is the difference between diffeomorphism and isomorphism?
QUESTION [6 upvotes]: What is the difference between diffeomorphisms and isomorphisms? I know isomorphisms already from my abstract algebra/group theory course, and now I'm studying analysis on (sub)manifolds, where this definition is new for me. Can anybody explain what the difference between the two is? Or in other words: how can I relate those two concepts? Thanks :)

REPLY [5 votes]: "Isomorphism" is a word with many different meanings, you shouldn't use it without context. In context of groups it means a group homomorphism that is bijective, in the context of topological spaces we might use it as another word for homeomorphism, which is a bijective continuous map with continuous inverse.
"Diffeomorphism" on the other hand is a very well defined word. It is a bijective continuously differentiable map with continuously differentiable inverse. To talk about "continuous differentiability" we at least need the structure of $C^1$-manifolds.