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How to prove $U \otimes Ind W = Ind(Res(U) \otimes W)$? where U is a representation of G and W is a rep of H, a subgroup of G. $Ind(W)$ is the induced rep and $Res(U)$ is the restrict rep.

Maybe I don't need a complete answer, so please just tell me, is the equal sign means being isomorphic? Are they the same meaning in rep theory, equal and isomorphic?

I got the answer, by both approaches: groups algebra and constructing isomorphic map. Thanks for all the very helpful comments and answer. I figured this is the wrong place to ask this question. This is the first time I ask here, I didn't know the rules, I apologize. I will come back after I know more math.

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How to prove $U \otimes Ind W = Ind(Res(U) \otimes W)$? where U is a representation of G and W is a rep of H, a subgroup of G. $Ind(W)$ is the induced rep and $Res(U)$ is the restrict rep.

Maybe I don't need a complete answer, so please just tell me, is the equal sign means being isomorphic? Are they the same meaning in rep theory, equal and isomorphic?

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