Isaacs, $\textit{Character Theory of Finite Groups}$, Theorem(1.10)

Let $V$ be an $A$-module. Then $V$ is completely reducible iff it is a sum of irreducible submodules.

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Pf:  Sufficiency $(\Leftarrow)$ 

  • $V=\sum V_{\alpha}$ and $W\subseteq V$
  • Choose $U\subseteq V$ maximal such that $W\cap U=0$ 
  • Prove $V=W\oplus U$

      Necessity $(\Rightarrow)$

  • Let $S$ be the sum of all the irreducible submodules of $V$
  • Prove by contradiction

      Remark: Every $A$-module contains an irreducible submodule.

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