Let $G$ be a finite group and let $((\rho_{n},V_{n}))_{n>0}$ a sequence of faithful representations of $G$ with $V_{n}$ a $\mathbb{C}$-vector space of dimension $n$.
Suppose that for all $g\in G$ and all positive integers $n$ and $m$, $(\rho_{mn},V_{nm})$ is the tensor product of $(\rho_{n},V_{n})$ and $(\rho_{m},V_{m})$.
Is every non trivial element of $G$ of order $2$? Are all elements of $\rho_{k}(g)$ complex numbers of modulus $1$ whatever the value of $k$?