Proposition 2 (Maschke)

Tag 0AU0, permanent. Cite it as [Bourbaki, Tag 0AU0].

Lie Groups and Lie Algebras, chapter V, Appendix 0. Complements on linear representations

Let A be a ring with unit element, E a left A-module, F a direct summand of E, G a group of finite order q, and $\rho$ a linear representation of G on E. Assume that q.1 is invertible in A and that F is stable under G. Then there exists a complement of F in E that is stable under G.

Let $p$ be a projection of $E$ onto $F$ . For all $x \in E$ , put

f(x) = q^{-1} \sum_{s \in G} \rho(s)^{-1} p(\rho(s)x).

We have

f(x) \in F

and

f(y) = y

for all

y \in F

, so

f

is a projection of

E

onto

F

. On the other hand, if

t \in G

\begin{align*} \rho(t)f(x) &= q^{-1} \sum_{s \in G} \rho(st^{-1})^{-1} p(\rho(s)x) \\ &= q^{-1} \sum_{s \in G} \rho(s)^{-1} p(\rho(st)x) \\ &= f(\rho(t)x). \end{align*}

Thus

f

commutes with

\rho(G)

, so

\ker(f)

is a complement of

F

stable under

G

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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Cites

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