In this question work over $\mathbb{C}$. Let $H$ be a subgroup of $G$. State Mackey's restriction formula, defining all the terms you use. Deduce Mackey's irreducibility criterion.

Let $G=\left\langle g, r: g^{m}=r^{2}=1, r g r^{-1}=g^{-1}\right\rangle$ (the dihedral group of order $2 m$ ) and let $H=\langle g\rangle$ (the cyclic subgroup of $G$ of order $m$ ). Write down the $m$ inequivalent irreducible characters $\chi_{k}(1 \leqslant k \leqslant m)$ of $H$. Determine the values of $k$ for which the induced character $\operatorname{Ind}_{H}^{G} \chi_{k}$ is irreducible.