Characters distinguish finite-dimensional complex semisimple representations

For any group, take the group-algebra image on the direct sum of two finite-dimensional complex semisimple representations. Its radical annihilates their simple summands and hence the faithful direct sum, so the image algebra is semisimple. Equality of traces on group elements gives equality on their linear span. Traces on primitive central blocks then recover every simple multiplicity, proving that equal characters imply isomorphism.