Intersection of two cyclic groups in $D_{n}$

Question

For the dihedral group $D_{n}=<\rho,\tau | \rho^{n}=\tau^{2}=id, \rho\tau=\tau\rho^{-1}>$

How can we prove that the intersection of $<\rho>$ and $<\tau>$ is the trivial group $\{e\}$, for all $n$? I know it is very obvious but I want to prove it formally.

Answer 1

Here is a simple argument.

If $\tau\in\langle \rho\rangle$, then $\tau$ commutes with $\rho$.

So $\tau\rho=\rho\tau=\tau\rho^{-1}$, hence $\rho=\rho^{-1}$, so $\rho^2=Id$ and $\rho$ has order $1$ or $2$. Hence $n=1,2$. If you assume $n\geq 3$ in your definition of $D_n$, we are done. If you allow $n=2$, then $D_n$ is the Klein group, and you can check your claim by hand.