Past exam of the introduction to representation theory course of the University of Oxford / 2023 / 2 / b / Solution

For each $h\in g^G\cap H$, count the solutions of $x^{-1}gx=h$. Once a solution $x_0$ is fixed, every solution is uniquely $c{x}_0$ with $c\in C_G(g)$, the centralizer of $g$. Thus each $h$ contributes $|C_G(g)|{\chi }_W(h)$ to the sum in part (a).

The orbit-stabilizer theorem for the conjugation action gives $|C_G(g)|=|G|/|g^G|$. The intersection $g^G\cap H$ is the disjoint union of the conjugacy classes $h_1^H,\dots ,h_{\ell }^H$, and the character ${\chi }_W$ is constant on each such conjugacy class. HenceIf the intersection is empty, the empty sum gives zero. This is the conjugacy-class formula for an induced character.