abstract algebra - Group isomorphism from subgroup of $U(n)times mathbb{Z}_n$ to $D_n$ the dihedral group of order 2n.

I have a group $G_n = U(n)\times \mathbb{Z}_n$ with the operation $(a,x)(b,y) = (ab,ay+x)$ and I have a subgroup $H_n = \{(a,b) \in G_n | a = \pm 1\}$ which I want to show is isomorphic to $D_n$ the dihedral group of order 2n.

I know for an isomorphism $\phi:H_n \to D_n$ I need $\phi(ab) = \phi(a)\phi(b)$ for $a,b \in H_n$. I know they have the same number of elements so that's good I suppose but I'm having trouble seeing how to preserve group operations with $\phi$.

I've tried to take $\phi: H_n \to \zeta_n$ the set of complex nth roots of unity under multiplication and conjugation (which is isomorphic to $D_n$ right?) because I thought it might be easier and I could then rely on the composition of isomorphisms being an isomorphism but I've not managed to find a working map $\phi$ and I'm very much starting to doubt that it's easier to go this route.

Can anybody point me in the right direction?

Answer

Indeed, $H_n$ and $D_n$ are isomorphic groups. First you may consider $D_n=\langle\tau^j\sigma^i\mid\;\tau^2=1=\sigma^n,\sigma\tau=\tau\sigma^{n-1}\rangle$, and define $\phi:H_n\to{D_n}$ such that $\phi(-1,0)=\tau$ and $\phi(1,1)=\sigma$. For instance you may take $\phi(a,b)=\tau^{\frac{1-a}{2}}\sigma^b$. The rest is straightforward.