show that $D_n =[e, a,a^2,...,a^n,b,ba,...,ba^{n-1}]$ forms a group

Question

Question : show that $D_n =[e, a,a^2,...,a^n,b,ba,...,ba^{n-1}]$ be a set of $2n$ elements. Define the product in $D_n$ by the relations $a^n =e, b^2=e$ and $ab = ba^{-1}$. Show that $D_n$ is a group.

As $D_n$ is a finite group, we can use composition table, but this is quite uncomfortable for me. Mention another ways...

Answer 1

Hint: Show that $D_n$ is the set of symmetries of a regular $n$-gon. Clearly such symmetries form a group.