TITLE: Homomorphisms and isomorphisms
QUESTION [5 upvotes]: If a homomorphism has a trivial kernel, is it an isomorphism?

REPLY [5 votes]: Recall: An isomorphism is a bijective homomorphism. 

We need more conditions on a homormorphism whose kernel is trivial to ensure it is an isomorphism:

A homomorphism whose kernel is trivial is injective, but not necessarily onto, hence not necessarily an isomorphism.

REPLY [3 votes]: It is not that actually. Even if you consider this identity map for example: $i:A_n \rightarrow S_n$ sending each $\tau$ to itself. $A_n$ is not isomorphic to $S_n$ right?
Ontoness is also necessary for an homomorphism to be an isomorphism. If you have an injective homomorphism which is not necessarily onto, then it is called an embedding.