The isomorphism from $S_3/\langle (123)\rangle$ to $\mathbb{Z}_2$. 
Suppose that $N=\{(123), (132), \operatorname{e}\}$ and $N$ is normal in $S_3$.  Show that the quotient group $S_3/N$ is isomorphic to $\mathbb{Z}_2$.

What mapping should I use?

First, by using Lagrange's theorem, we know the index $[S_3:N]=|S|/3=6/3=2$, so that will be the order of the quotient group...
Now the mapping between this and $\mathbb{Z}_2$ is trivial because, under isomorphisms, there exists only one group of order 2. I will let you realize that, just think about it, if you have a group of only two elements, not a lot of things can happen...
