This is a standard result that any beginner in the study of the representation theory would be aware of. I merely present restate the theorem a little(essentially I don’t) and prove it.
Let be representations of a group . If is a map, then the following are true assuming that and are not isomorphic:
1)If both are irreducible, then is either an isomorphism or the zero map.
2)If only is irreducible, then is either injective or the zero map.
3)If only is irreducible, then is either surjective or the zero map
If are isomorphic as representations, then is a scalar multiplication map.
Let be the respective representations. Here, are vector spaces over an algebraically closed field.
We tackle the first part of the theorem.
Consider the kernel of , say , which is set of all such that . Since is a G-linear map, implies that if , then . Hence, is stable under the action of which makes it a sub-representation. Since is irreducible, this means that either must be zero or must be the zero map. This proves that is injective.