Equivalent definitions to $D_{n}$

Question

I am looking for equivalent definitions for $D_{n}$.

I have this: The Dihedral group of degree $2n$ ($D_{n}$) is the group of symmetries of a regular polygon of $n$ sides. But, when I am trying to work with it, this definition is not much useful.

I saw a definition of it using two elements $a$ and $b$ of order $n$ and $2$, respectively. Can someone explain this construction in detail, and how I can prove that those definitions are equivalent?

Answer 1

One definition is that $D_n$ of order $2n$ is the group given by the presentation

$$\langle a,b\mid a^2, b^n, ab=b^{-1}a\rangle.$$

Here $a$ corresponds to a flip of an $n$-gon, whereas $b$ is a rotation by $\frac{2\pi}{n}$.