Dihedral groups and reflections

Question

Let $\rho\in D_{10} $ be a rotational symmetry in $\frac{2\pi}{5}$ radians non-clockwise,and let $\epsilon\in D_{10}$ be a reflection symmetry (related to the X-axis).Prove that every $\epsilon\rho^i\in D_{10}$ is a reflection symmetry.

For $i=5$,I know that $\rho^5=e=\epsilon^2$,so $\epsilon\rho^5=\epsilon$.But I don't know what can I say about $1\le i \le 4$... Any ideas/hints?

Answer 1

Note that $\rho$ and $\epsilon$ in your question can be related by $\rho^{i} \epsilon \rho^{i} = \epsilon$. You can convince yourself about this by drawing a figure.

Therefore, $(\epsilon \rho^i)^2 = \epsilon (\rho^i \epsilon) \rho^i = \epsilon (\epsilon \rho^{-i}) \rho^i = e$.

So $\epsilon \rho^i$ is a symmetry of pentagon which squares to the identity, it must be a reflection then.

Answer 2

All the rotations are of the form $\rho^j$ so if $\epsilon \rho^i$ were a rotation we would have $\epsilon \rho^i = \rho^j$, whence $\epsilon= \rho^{j-i}$. But $\epsilon$ isn't a rotation, so this is a contradiction. Hence $\epsilon \rho^i$ is a reflection.