I have this exercise:
Let $f$ be a function such that $$f(x,y)=(-x,y)$$ and $h$ the rotation of the plane respect to the origin in an angle of $\frac{2\pi}{n}$. Let $G$ be such that $$G=\lbrace f^ih^j\mid i=0,1;j=0,1,2,...,(n-1)\rbrace$$ and the usual product of functions $*$. If $G$ is a dihedral group prove that:
a) If $n$ is odd and $a\in G$ such that $a*b=b*a$ for all $b\in G$, then $a=e$ ($e$ its the identity of the group).
b) If $n$ is even, prove that exists $a\in G$, $a\neq e$, such that $a*b=b*a$, for all $b\in G$.
c) If $n$ is even, find every element $a\in G$ such that $a*b=b*a$, for all $b\in G$.
I don't know how to do it. Help please.
Hint
Since $G$ is a dihedral group, it satisfies the relations $$f^2=h^n=1,hf=fh^{-1}.$$
For how to get these relations, you may refer here.
Let $z\in G$ such that $zg=gz$ for all $g\in G$.
Then in particular you have $$zh=hz \text{ and } zf=fz.$$ Write $z=f^ih^j$.
By solving the first equation, you will get $i=0$.
And by solving the second equation, you will get $j=0$ when $n$ is odd, and $j=0,n/2$ when $n$ is even.
Hence if $n$ is odd, $z$ must be $e$, and if $n$ is even, $z$ is $e$ or $h^{n/2}$.