I am asked to show that the subgroup $G$ of the Mobius group $M$, generated by $f(z)=e^{2\pi I/n}z$ and $g(z)=\frac{1}{z}$ is isomorphic to $D_{2n}$. I considered the mapping

$$h: G \to D_{2n} \text{ with } h(g)=s, h(f)=r$$

Now I thought I would need to show first that it is well defined, and second that it is a homomorphism and bijection. However I can't quite figure out what it would mean to show that this map is well defined. I can certainly show that $gf^k=f^{-k}g$, and $sr^k=r^{-k}s$ is true by definition of the dihedral group, but I don't see if this does or does not constitute a 'proof' that the map is well defined?

Also, I just wanted to verify that the action of a homomorphism is completely specified by what it does to the generating elements of some group. From the definition of homomorphism, this seems to be obviously true, however there is something that keeps preventing me from being satisfied just with this descripton when it comes to actually doing questions. For example, in the above, I wanted to specify $h(fg)$ etc. Perhaps this is a different matter, because when II a specifying the action of $h$ above I do not yet know if it a homomorphism or not?

Thank you in advance!


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