每日定理5

Posted 2021-02-17 zhengtao1992

Isaacs, $ extit{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.

Pf:  Sufficiency $(Leftarrow)$ 

• $V=sum V_{alpha}$ and $Wsubseteq V$
• Choose $Usubseteq V$ maximal such that $Wcap U=0$ 
• Prove $V=Woplus U$

      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.