Normal subgroups and functions from cosets to cosets Let $K ⊂ G$ be a normal subgroup and let $G/K$ be the set of left cosets of $K$. 
Consider
the function
$G/K × G/K → G/K$ given by $(g1 · K, g2 · K) → (g1 · g2) · K$.
(a) Prove this function is well-defined.
(b) Prove this function makes $G/K$ a group.
What does it mean for the function to be well defined in this case?

The function has cosets as domain, but the definition is written using representatives of the cosets ($g_1$ and $g_2$ in this case). When writing it that way, there is always the danger of the function taking different values on the same cosets if you choose a different representative. Such a function would not be well-defined.
Showing that the function is well-defined means showing that choosing different representatives for the same cosets as input to the function gives the same coset as output.
To show what I mean, here is a function which is not well-defined. Take $f:\Bbb Q\to \Bbb Z$ defined by $f(a/b) = a+b$. Then $f(1/1) = 2$, but $f(2/2) = 4$. The fractions $\frac11$ and $\frac22$ represent the same rational number, but give different function values. Thus $f$ is not well-defined.
