Past exam of the algebra course of the University of Oxford / 2010 / A / 3 / b / ii

Solution

Let

w = \sum_{g \in G} g

. This is nonzero since its basis coefficients are all one, and

g w = w

for every

g \in G

. Hence

k w

is a one-dimensional left ideal of

k G

If the regular module is semisimple,

k w

π : k G \to k w

be the corresponding

k G

-linear projection, so

π (w) = w

. Write

π (1) = c w

for

c \in k

. Then

w = π (w) = \sum_{g \in G} π (g) = \sum_{g \in G} g π (1) = c ∣ G ∣ w .

Because

w \neq = 0

c ∣ G ∣ = 1

k

. Thus

∣ G ∣ \neq = 0 in k .

Together with the Maschke theorem, this gives the exact semisimple algebra criterion for a group algebra.

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