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Let $G$ be a finite group and $\mathbb{C}$ be the complex field. $L,M,N$ are finitely generated $\mathbb{C}G$-modules (Where $\mathbb{C}G$ denotes a group ring).

Show if $$0\rightarrow L \rightarrow M \rightarrow N\rightarrow 0$$ is a short exact sequence then it splits.

Maddy
  • 1,875
  • This is textbook representation theory - namely Maschke's theorem. The typical proof involves Weyl's unitary trick: put an inner product on $M$, average it over $G$ to make it invariant, then define the orthogonal complement of $L$, which you can verify is also invariant, and so $M$ is a direct sum. – anon May 27 '18 at 15:25

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