I'm not much familiar with this topic, but nontheless I have a problem understanding the context. Following this link Second Isomorphism Theorem, they state that $HN/N$ is a group under left coset multiplication. They then define the map $\phi(h)=hN$ (a map from $H$ to $HN/N$) where $N$ is a normal subgroup of a larger group $G$ and $H$ is a subgroup of $G$. Furthermore they state $\phi$ is clearly a homomorphism. How is that? How do I understand the algebraic operation between these elements?
For instance, we would have to have $\phi(h\cdot h^{-1})=\phi(e)=N=\phi(h)\cdot \phi(h)^{-1}=hN\cdot (hN)^{-1}$. So as far as I undertand, the elements of $HN/N$ are the cosets of $HN$, right? How do you take the inverse of a coset?
Best Answer
I think the thing you are struggling with is the definition of the group operation on $HN/N$, or more generally on $G/N$. Here, multiplication is defined as: $$gNhN=ghN$$ which makes sense, as by normality of $N$ we have $gNhN=g(hNh^{-1})hN=ghNN=ghN$. Therefore, inverses are as follows: $$(gN)^{-1}=g^{-1}N$$
As the map you are considering is defined by $\phi(h)=hN$, we clearly have: $$\phi(g)\phi(h)=gNhN=ghN=\phi(gh)$$ as required for homomorphisms.