每日定理6

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.

Pf:

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

Isaacs, $\textit{Character Theory of Finite Groups}$, Lemma(1.13)

Let $V=\sum\cdot W_i$ be a direct sum of $A$-modules with $W_i$ irreducible for all $i$. Let $M$ be any irreducible $A$-module. Then

1. $M(V)$ is an $E_A(V)$-submodule of $V$;
2. $M(V)=\sum\{W_i|W_i\cong M\}$;
3. The number $n_M(V)$ of $W_i$ which are isomorphic to $M$ is an invariant of $V$, independent of the direct sum decomposition.

Pf: 1.

• Let $\vartheta\in E_A(V)$, $W\subseteq V$ and $W\cong M$ 
• $W\vartheta=0$ or $W\vartheta\cong W\cong M$

     2.  Let $W\subseteq V$ and $W\cong M$

• Let $\pi_i$ be the projection map of $V$ onto $W_i$
• $\pi_j(W)=0$ or $\pi_j(W)=W_j\cong M$
• $W\subseteq\sum_j\pi_j(W)\subseteq\sum\{W_i|W_i\cong M\}$

     3.  By 2

• $dim M(V)=n_M(V)\cdot dim(M)$

Remark:

• $E_A(V)$ is an $F$-algebra.