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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2019

Paper 3

Let $D$ represent $H\backslash G/H$. Combining <Frobenius reciprocity> with the <Mackey's theorem> gives
$$
\begin{aligned}
\left\langle1_H\mathbin{\uparrow}^G,
1_H\mathbin{\uparrow}^G\right\rangle_G
&=\left\langle1_H,
\operatorname{Res}_H^G(1_H\mathbin{\uparrow}^G)\right\rangle_H\\
&=\sum_{x\in D}
\left\langle1_H,
1_{H\cap xHx^{-1}}\mathbin{\uparrow}^H\right\rangle_H\\
&=\sum_{x\in D}
\left\langle1_{H\cap xHx^{-1}},
1_{H\cap xHx^{-1}}\right\rangle_{H\cap xHx^{-1}}\\
&=|D|.
\end{aligned}
$$
Thus
$$
\boxed{
\left\langle1_H\mathbin{\uparrow}^G,
1_H\mathbin{\uparrow}^G\right\rangle_G
=|H\backslash G/H|.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 19I / b / ii / Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 19I / b / ii / Solution

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Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 19I / b / ii / Solution

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