Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 4 / 4 / e

Table of contents
- Solution e

Solution

If

N_{G} (D) \leq H \leq G

, then

N_{H} (D) = N_{G} (D) = N

and

C_{H} (D) = C_{G} (D) = C

. Applying part (d) to

G

and to

H

therefore gives the same target ideal

J_{D} = Tr_{D}^{N} (k C) .

Compose the two bijections. A block idempotent

e

of

G

with defect

D

corresponds to the unique block idempotent

f

of

H

with defect

D

satisfying

Br_{D}^{G} (e) = Br_{D}^{H} (f) .

This proves the requested form of Brauer first main theorem, including uniqueness and the equality of the Brauer images.

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