Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 4 / 1 / i / Solution

Use the defining property of the Frobenius complement: $H\cap H^x=\{1\}$ whenever $x\notin H$. The induced character formula, extended linearly to any class function, isFor $h\in H\setminus \{1\}$, a summand requires $h\in H\cap xH{x}^{-1}$. The Frobenius intersection property forces $x\in H$. All $|H|$ remaining terms equal $\theta (h)$ since $\theta$ is a class function on $H$, so ${\theta }^G(h)=\theta (h)$. At the identity,Together these giveThe zero identity value is essential: at nonidentity elements the intersection argument already gives the equality, whereas induction multiplies the identity value by the subgroup index.