Let $G$ be a finite group with representations $\rho_1, \rho_2:G\rightarrow GL(V)$. According to the definition of representation isomorphisms, $\rho_1$ and $\rho_2$ are isomorphic if there exists a function $\phi:V\rightarrow V$ such that $\phi(gv)=g\phi(v)$ for all $g\in G$, $v\in V$.
Why would choosing the trivial isomorphism $\phi(v)=v$ for all $v\in V$ not show that all representations from $G$ into $V$ are isomorphic? Then $\phi(gv)=gv=g\phi(v)$? Obviously not all group representations into the same vector space are isomorphic, so what is the flaw in the reasoning here?
Thanks so much.