Non-trivial isomorphism between the dihedral group to itself.

Question

I want to find a non-trivial isomorphism between the dihedral group $D_n$ and itself. Non-trivial means that the isomorphism won't be the identity.

I looked at the group $D_n$ as the set of the following elements:

$$\{e, \sigma, \sigma^2, ...., \sigma^{n-1} , \rho, \rho \sigma, \rho \sigma^2 ..., \rho \sigma^{n-1}\},$$

where $\sigma$ is a $\frac{2\pi}{n}$ rotation (clockwise) and $\rho$ is reflection through the vertical line.

What I thought about is the following function: $f(\sigma^k) = \sigma^{-k}$ and $f(\rho\sigma^{-k}) = \rho\sigma^{-k}$

It seems to that it is legit.

Am I correct here?

Answer 1

An isomorphism from a group $G$ to the same group $G$ is called an automorphism.

See this YouTube video describing the group of automorphisms of the dihedral group. (Yes, they form a group! The operation is composition of functions.)