How to solve the problem from Topics In Algebra Herstein?

Question

Let $G$ be the dihedral group defined as the set of all formal symbols $x^iy^j$, $i=0,1$, $j=0,1,\ldots,n-1$, where $x^2=e$, $y^n=e$, $xy=y^{-1}x$. Prove

1. The subgroup $N=\{e,y,y^2,\ldots,y^{n-1}\}$ is normal in $G$.
2. That $G/N\approx W$, where $W=\{1,-1\}$ is the group under the multiplication of the real numbers.

I have solved (a) part .In part (b) we need to define homomorphic function.What i was thinking that after defining a homomorphic function if we prove that N is kernel then we are done .But i am unable to find a homomorphic function.

Answer 1

Define$$\begin{array}{rccc}\varphi\colon&G&\longrightarrow&\{1,-1\}\\&x^iy^j&\mapsto&(-1)^i.\end{array}$$Prove that it is a group homomorphism. It is clear that $\ker\varphi=N$.

Answer 2

Hint - Define a similar map that Jose Carlos Santos has suggested in his answer from $G/N$ to $W$ . Since number of elements in $G$ is equal to $2n$ and number of elements in N is n this implies number of elements in $G/N$ is $2$ which is equal to number of elements in $W$.