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Suppose two groups have the same character table of complex representations. Also, all the entries in this character table have absolute value at most $1$. Does this imply that the two groups are isomorphic?

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    Are those finite groups you're asking about?2012-06-15
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    Do character tables even make sense if the groups are not finite?2012-06-15
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    there is no information about groups if they are finite or not2012-06-15
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    William: they might still make sense for compact groups.2012-06-15
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    You can think of the character table as a pairing between two sets, the irreducible finite-dimensional representations and the conjugacy classes, that gives a complex number. For two arbitrary groups it still makes sense to ask for a bijection between the conjugacy classes and irreducible representations of the two groups that allows us to identify the two pairings.2012-06-15

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