Define $D_n = {1, r, r^2,..., r^{n-1}, t, tr, tr^2,... tr^{n-1}}$. Prove that $D_n$ is a group.
I know that I have to first show that $D_n$ is closed under the operation first, i.e. that $rt=tr^{n-1} = tr^{-1}$, is this something to be proved through induction?
Further, $D_n$ is the dihedral group, order 2n, and is clearly finite. Though the set seemingly has clear associativity and invertibility, the identity portion is tricky to me.