I'm having problems understanding the excersice:
E) Define $D_n$ as the group of symmetries of a regular n-gon.
Name the vertices $V=\{V_0,V_1,…,V_{n-1}\}$ so that $$V_{k}=\exp({i\cdot\dfrac{2\pi k}{n})}$$
Let $S$ be the symmetry along the x-axis, and $R$ the rotation by $\dfrac{2\pi}{n}$.
Show that $\{R,S\}$ generate $D_n$.
I tried to define something like $$R:\exp(i\cdot\dfrac{2\pi k}{n})\mapsto \exp(i\cdot\dfrac{2\pi (k+1)}{n}) $$ $$S:\exp(i\cdot\dfrac{2\pi k}{n})\mapsto \exp(i\cdot\dfrac{2\pi (-k)}{n}))$$
but these are not binary operation so $(V,R) ; (V,S)$ are not groups. To me it's clear that there are n rotations and n reflection so there are 2n symmetries on the n-gon; also fixing one reflection we can get the others by some combination of $r,s$.
I tried to solve the problem by defining $R,S$ as permutation of vertices and the vertices as coordinates in a lattice and it solves nicely, but I don't seem to understand this way of approaching the definitions.
Best Answer
Your definitions of $\ R\ $ and $\ S\ $ are basically correct, except that they need to be operations on the whole complex plane: \begin{eqnarray} \ R\left(z\right) &=& \exp\left(\dfrac{2\pi i}{n}\right)z\\ S\left(z\right) &=& \overline{z}\ , \end{eqnarray}
and you seem to be a little confused about what the group operation of the group of symmetries is. The group operation is composition. That is, $\ R\,S\ $ is the symmetry obtained by applying $\ S\ $ then $\ R\ $ $$ R\,S\left(z\right)= R\left(S\left(z\right)\right)\ .$$ With these definitions you should be able to show that $\ R^k\hspace{-0.2em}\left(V_j\right) = V_{j+k\hspace{-0.3em}\pmod{\hspace{-0.2em}n}}\ $, and $\ S\left(V_j\right)=V_{n-j}\ $, and thence that any symmetry can be obtained as some combination of $\ S\ $ and powers of $\ R\ $.