Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 19I / a

Table of contents
- i a
  - Solution i
- ii a
  - Solution ii

i

Solution

For an

H

-module

W

,

Ind_{H}^{G} W = C [G] \otimes_{C [H]} W .

The action is left multiplication on the first factor:

g_{0} \cdot (g \otimes w) = g_{0} g \otimes w .

It is well defined because the tensor relation

g h \otimes w = g \otimes h w

is preserved by left multiplication.

ii

Solution

If the

H

-classes in

H \cap C (g)

have representatives

x_{i}

, the induced-character formula is

(Ind_{H}^{G} ψ) (g) = ∣ C_{G} (g) ∣ i = 1 \sum s \frac{ψ ( x _{i} )}{∣ C _{H} ( x _{i} ) ∣} .

It follows by grouping the standard sum over

x \in G

satisfying

x^{- 1} gx \in H

.

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