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Dec 2, 2010 at 19:43 comment added user6976 The groups are residually finite so they have lots of finite dimensional unitary representations. I have answered the updated question also. But it is much less trivial than the original question.
Dec 2, 2010 at 19:03 comment added Theo Johnson-Freyd Well, if you have two nonisomorphic groups neither of which has finite-dimensional matrix representations, then certainly the answer to your updated question is yes!
Dec 2, 2010 at 12:25 history edited user6976 CC BY-SA 2.5
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Dec 2, 2010 at 10:47 history answered user6976 CC BY-SA 2.5