= 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.
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