For $n \ge 3$, describe the elements of $D_n$. How many elements does $D_n$ have?

Question

I am trying to refresh my knowledge of modern algebra, and have come across a question in Gillian 8th ed. that I don't quite remember how to prove.

The question is:

For $n \geq 3$, describe the elements of $D_n$. How many elements does $D_n$ have?

I want to prove that $D_n$ is a complete set (I know it has $2n$ elements). I assume the general structure would be to prove that the list of symmetries (we'll call it $L$) is contained in the set of symmetries (we'll call it $\Sigma$), and then the other way, $\Sigma \subseteq L$.

If someone could point me in the right direction it would be much appreciated!

Answer 1

Hint Label the vertices $1,2,3,..,n$.

An element in $D_n$ will take $1$ to some vertex $k$ and $2$ to one of the two neighbours of $k$. Show that this completely determines the symmetry.