Let $G$ be a finite group and $\chi$ be an irreducible character of $G$ (characteristic zero algebraically closed base field). I f$H$ is the kernel of $\chi$ then the irreducible representations of $G/H$ are exactly all the irreducible constituents of all tensor powers $\chi^n$.

1. Do you know any reference for this theorem?

2. Is it also working in positive characteristic?

3. Is it also working for some infinite groups? (maybe some special classes: reductive, Lie type, etc)

Thank you very much!

Let $G$ be a finite group and $\chi$ be an irreducible character of $G$ (characteristic zero algebraically closed base field). If $H$ is the kernel of $\chi$ then the irreducible representations of $G/H$ are exactly all the irreducible constituents of all tensor powers $\chi^n$.

1. Do you know any reference for this theorem?

2. Is it also working in positive characteristic?

3. Is it also working for some infinite groups? (maybe some special classes: reductive, Lie type, etc)

Thank you very much!