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 Homomorphism theorem (link) and the first thing the proof does is define a map and find it if its 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 its injective.

What have I understood wrongly?

Soham

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.

Suppose that I try to define a map $f$ from $\Bbb Q$, the set of rational numbers, to $\Bbb Z$, the set of integers by setting $f\left(\frac{a}b\right)=a$; what is $f(1)$?

$1=\frac11$, so $f(1)=f\left(\frac11\right)=1$.

But wait! $1=\frac22$, so $f(1)=f\left(\frac22\right)=2$.

And $1=\frac{100}{100}$, so $f(1)=100$.

Obviously this doesn’t work: by my ‘definition’ $f(1)$ could be any integer at all. In other words, my supposed definition doesn’t actually define anything: $f(1)$ depends on which representation of $1$ as a fraction of two integers I use, and nothing in the ‘definition’ requires me to pick one particular representation. This supposed function is not well-defined.

On the other hand, every rational number $q$ can be uniquely represented in the form $\frac{a}b$ where $\gcd(a,b)=1$ and $b>0$. Had I defined $f(q)$ to be the numberator $a$ of this specific representation, $f$ would have been a genuine function: it would have been well-defined.

Checking that a mathematical object is well-defined is really just checking that it is defined: that the purported definition actually does unambiguously specify the object.