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 $Hom_A(W,V)$.

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Pf: Let $0\neq\varphi\in Hom_A(V,W)$,

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