Proposition 9

Let S be a simple submodule of V and T the simple submodule $\varphi (S)$ of M. If $V_S$ denotes the isotypical component of V of type S and $M_T$ the isotypical component of M of type T, then we have $\varphi (V_S) = M_T$.

Every simple submodule $S'$ of V distinct from S satisfies $S'\cap S = 0$. It is therefore isomorphic to S if and only if $\varphi (S')$ is isomorphic to T (Lemma 2). Now, $V_S$ is the sum of the simple submodules of V isomorphic to S, and $M_T$ is the sum of the simple submodules of M isomorphic to T. Proposition 9 therefore follows immediately from Propositions 7 and 8.