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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

On the orbit $G/H$, an element $g\in G$ fixes the coset $xH$ exactly when
$$
gxH=xH
\quad\Longleftrightarrow\quad
x^{-1}gx\in H.
$$
The <character of a representation> of the corresponding <permutation representation> therefore is
$$
\boxed{
\chi_{\mathbb C(G/H)}(g)
=|(G/H)^g|
=\frac1{|H|}
|\{x\in G:x^{-1}gx\in H\}|
=(1_H\mathbin{\uparrow}^G)(g).}
$$
This is the <permutation character of a coset action>.

Solution

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

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

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

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