Isaacs, $\textit{Character Theory of Finite Groups}$, Lemma(1.11)

Let $V$ be an $A$-module and suppose $V=\sum V_{\alpha}$ where the $V_\alpha$ are irreducible submodules. Then $V$ is the direct sum of some of the $V_\alpha$'s.

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Pf:

  • Choose $W$ maximal with the property that $W$ is the direct sum of some $V_\alpha$'s
  • Prove by contradiction   
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