This happens if and only if $V$ U$is expressible as a sum of pairwise non-isomorphic irreducible$G$-modules. If, for example$V U \cong U V \oplus U$V$ for an irreducible $G$-module $U,$ V,$then the natural invariant submodule $ \{(u,0): u {(v,0): v \in U V \}$ has at least two complements: one is the natural choice $\{(0,u): u \{(0,v): v \in U V \}.$ Another is $\{(u,u): u \{(v,v): v \in U V \}.$ On the other hand, if$V U \cong \bigoplus_{i=1}^{n} U_i,$V_i,$ where $U_i V_i \not \cong U_j$ V_j$for$i \neq j,$then the only$G$-submodules of$V$U$ are of the form $U_I V_I = \bigoplus_{i \in I} U_i$ V_i$for a subset$I$of $\{1,2,\ldots ,n\},$ and each such$G$-submodule$U_I$V_I$ has the unique complement $U_{I^{\prime}},$ V_{I^{\prime}},$where $I^{\prime} = \{ 1,2,\ldots ,n \} \backslash I.$ 4 latex again This happens if and only if$V$is expressible as a sum of pairwise non-isomorphic irreducible$G$-modules. If, for example$V \equiv cong U \oplus U$for an irreducible$G$-module$U,$then the natural invariant submodule $ \{(u,0): u \in U \}$ has at least two complements: one is the natural choice $\{(0,u): u \in U \}.$ Another is $\{(u,u): u \in U \}.$ On the other hand, if$V \equiv cong \bigoplus_{i=1}^{n} U_i,$where$U_i \not \equiv cong U_j$for$i \neq j,$then the only$G$-submodules of$V$are of the form$U_I = \bigoplus_{i \in I} U_i$for a subset$I$of ${1,2,\ldots \{1,2,\ldots ,n},$n\},$ and each such $G$-submodule $U_I$ has the unique complement $U_{I^{\prime}},$ where $I^{\prime} = \{ 1,2,\ldots ,n \} \backslash I.$
This happens if and only if $V$ is expressible as a sum of pairwise non-isomorphic irreducible $G$-modules. If, for example $V \equiv U \cong oplus U$ for an irreducible $G$-module $U,$ then the natural invariant submodule $\{(u,0): u \in U \}$ has at least two complements: one is the natural choice $\{(0,u): u \in U \}.$ Another is $\{(u,u): u \in U \}.$ On the other hand, if $V \equiv \bigoplus_{i=1}^{n} U_i,$ where $U_i \not \equiv U_j$ for $i \neq j,$ then the only $G$-submodules of $V$ are of the form $U_I = \bigoplus_{i \in I} U_i$ for a subset $I$ of ${1,2,\ldots ,n},$ and each such $G$-submodule $U_I$ has the unique complement $U_{I^{\prime}},$ where $I^{\prime} = \{ 1,2,\ldots ,n \} \backslash I.$