# Elements of $D_{2n}$ in terms of isometries

In course of studying Dihedral Group I'm having trouble to get what exactly the elements of $D_{2n}$ are. According to the Dummit-Foote texts

For each $n∈\mathbb Z^+,n≥3$ let $D_{2n}$ be the set of symmetries of a regular $n$-gon,where a symmetry is any rigid motion of the $n$-gon which can be effected by taking a copy of the $n$-gon,moving this copy in any fashion in $3$-space and then placing the copy back on the original $n$-gon so it exactly covers it.

I know that rigid motion of a plane shape is actually an isometry from the shape (considered as a metric subspace of $\mathbb R^2$) to $\mathbb R^2$. Keeping this in mind is it justified to say $D_{2n}$ as the set of all isometries from the shape to $\mathbb R^2$ such that the images are the shape itself (or simply as the set of all isometries from the shape onto itself)?

Yes, you can think of $D_{2n}$ as isometries of $\mathbb R^2$ that preserve the $n$-gon. In fact, this will be one of your first examples of a representation of $D_{2n}$, so keep it in mind because it will probably pop up later in your studies.