I teaching myself character theory, but I don't understand a problem statement from Dummit and Foote, Exercise 18.3.4.
Prove that if $N$ is any irreducible $\mathbb{C}G$-module, and $M=N\oplus N$, then $M$ has infinitely many direct sum decompositions into two copies of $N$.
If $M$ is a direct sum decomposition into two copies of $N$, doesn't that mean $M=N\oplus N$? The only way I can make sense if this is that it means to show that $M$ can be decomposed in infinitely many ways as $M=N_1\oplus N_2$ where $N_1$ and $N_2$ are isomorphic to $N$?
In the special case where $G$ is trivial, $N\simeq\mathbb{C}$, so $M\simeq\mathbb{C}^2$. Then there are infinitely many ways to write $M$ as a direct sum, with each summand isomorphic to $\mathbb{C}$, hence to $N$, namely $$ M=\langle [1,0]^T\rangle\oplus\langle [a,1]^T\rangle $$ by letting $a$ range over $\mathbb{C}$. These decompositions are distinct since $\langle [a,1]^T\rangle\neq\langle[b,1]^T\rangle$ unless $a=b$.
I think the idea for arbitrary finite group $G$ is to somehow choose $\mathbb{C}G$-bases which generate infinitely many distinct submodules isomorphic to $N$? But I'm not sure how to do that.