elements of order two in $D_{10}$

Question

Which elements have order two in $D_{10}$?

In $D_{10}$ there are $10$ elements, five of which are rotations and five reflections. Let $\rho = (1\hspace{1mm}2 \ldots 5)$ and $\tau = (1)(2\hspace{1mm}5)(3\hspace{1mm}4)$. Powers of $\rho$ have orders $5$, while $\tau$ has order $2$. My question is: do all of the other reflections $\tau\rho,\tau\rho^2,\tau\rho^3,\tau\rho^4$ have orders $2$? I was trying to use the identity: $\rho^i\tau=\tau\rho^{5-i}$ and this shows that all of them have order $2$, but I would like for someone to point out if this is ok or not.

Answer 1

Yes, the reflections $\tau\rho,\tau\rho^2,\tau\rho^3,\tau\rho^4$ have order two. For $(\tau\rho^i)^2=\tau\rho^i\tau \rho^i==\tau\rho^i\rho^{5-i}\tau=\tau\rho^5\tau=\tau\tau=e$.

Answer 2

In the dihedral group, all of the reflections have order $\color{blue}2$,

and that can be shown using the formula $\rho^i\tau=\tau\rho^{-i}$:

$(\tau\rho^i)^\color{blue}2=\tau\rho^i\tau\rho^i=\tau\tau\rho^{-i}\rho^i=\tau^2$ is the identity.