= 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 $cx_0$ 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> $h_1^H,\ldots,h_\ell^H$, and the <character> $\chi_W$ is constant on each such <conjugacy class>. Hence
$$
\chi_V(g)=\frac{|C_G(g)|}{|H|}\sum_{h\in g^G\cap H}\chi_W(h)
=\boxed{\frac{|G|}{|H|}\sum_{i=1}^{\ell}\frac{|h_i^H|}{|g^G|}\chi_W(h_i)}.
$$
If the intersection is empty, the empty sum gives zero. This is the <conjugacy-class formula for an induced character>.
Back to article page
