In Group theory proofs what is meant by "well defined" What is exactly meant or required for a mapping to be well defined? I was reading the First Isomorphism theorem (link), and the first thing the proof does is define a map and find out if it's well defined.
Intuitively it makes sense, but what are the requirements for a map to be well defined? For example in the link given, I understand they show one-one relationship as being well defined and later on they again prove it's injective.
What have I understood wrongly?

One interesting observation is that "well-defined" is basically the converse of (so closely related to) "one-to-one". That is:

We say that $\varphi$ is well-defined if $g=h$ implies that $\varphi(g)=\varphi(h)$.
We say that $\varphi$ is one-to-one if $\varphi(g)=\varphi(h)$ implies that $g=h$.

Thus, if we're trying to prove $\varphi$ is a one-to-one homomorphism (or perhaps even an isomorphism), we can sometimes get that $g=h$ if and only if $\varphi(g)=\varphi(h)$, using double implications the whole way, so that we simultanously prove that $\varphi$ is both well-defined and one-to-one, rather than dealing with them in two separate steps. That then leaves only showing homomorphism (and onto, if we're trying to prove isomorphism). It isn't always so simple--occasionally, we'll need a slick trick to show one-to-one, which doesn't neatly lend itself to reversal and showing well-defined. Still, it's a nice thing to keep in mind as a possibility.
