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Permutation representation

For a subgroup $H\leq G$, the permutation character of the <coset action> on $G/H$ is the induced trivial character
$$
\pi_{G/H}=1_H\mathbin{\uparrow}^G,
$$
and
$$
\pi_{G/H}(g)
=|\{xH:gxH=xH\}|
=\frac1{|H|}|\{x\in G:x^{-1}gx\in H\}|.
$$

Permutation character of a coset action

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 19I / b / ii / Solution

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

Permutation character of a coset action

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Permutation character of a coset action

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