I'm reading Ash's Basic Abstract Algebra. Here:
I'm a little bit confused. When we restrict the map $\pi: G\to G/N$, we get the map $\pi_0:H\to H/(H\cap N)$, right? Assuming this is true, I don't see how the image of $\pi_0$ is what is given in there.
In $\pi_0$, $H$ is partitioned in the classes $H\cap N, h_1(H \cap N )\dots$, I think we would need a partition such as $N,h_1 N, h_2 N, \dots$, in this case I guess we should have $\pi_0:H\to H/N$ but I don't know if this makes sense because $N$ could have elements not in $H$.