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Questions about the branch of algebra that deals with groups.

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When do two maps between groups give the same map between representation rings?

If I have a homomorphism $f: G\to H$ of groups, I get a homomorphism $R(f)\colon R(H)\to R(G)$ of representation rings in the opposite direction, by composition. Given two homomorphisms $f_1,f_2\colon …
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