What’s going on when I restrict a homomorphism to a smaller quotient group?

Question

I have $N\lhd G$, $K\lhd G$ and $N\le K$. Let $f:G\rightarrow H$ be a homomorphism with kernel $K$. I want to write $f$ as a composition $h\circ\pi$ where $\pi:G\rightarrow G/N$ is the projection and $h:G/N\rightarrow H$ is a homomorphism.

To get $h$, I can just map $gN$ to $f(g)$ because if $gN=rN$ then $f(g)=f(r)$. What’s actually happening in the last sentence abstractly in terms of the homomorphisms/projections involved?

We needed to use that $N\le K$ to say this.

Answer 1

What you wrote in your sentence "To get $h$" is an exactly correct abstract description of the homomorphism. If what you wanted was to break up the (exactly correct) implication "if $gN=rN$ then $f(g)=f(r)$" into a few more steps of set theoretic logic then you could write something like this:

If $gN=rN$ then $gN=rN = gN \cap rN \subset gK \cap rK$. It follows that $gK \cap rK \ne \emptyset$, and therefore $gK=rK$, so $f(g)=f(r)$.