每日定理1

Isaacs, $\textit{Character Theory of Finite Groups}$, Lemma(1.5)

If $V$ and $W$ are irreducible $A$-modules, then every nonzero element of $Hom_A(V,W)$ has an inverse in $Hom_A(W,V)$.

Pf: Let $0\neq\varphi\in Hom_A(V,W)$,

• $\ker\varphi$ is a submodule of $V$
• $im\varphi$ is a submodule of $W$