Sufficiently large implies character-separating

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This fact is related to: linear representation theory
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Statement

Let G be a finite group and k a field whose characteristic does not divide the order of G. Then, if k is a sufficiently large field for G (that is, k contains all the m^{th} roots of unity where m is the exponent of G), then k is a character-separating field for G.

By k is character-separating for G, we mean that given two distinct conjugacy classes c_1 and c_2 of G and elements g_i \in c_i, there exists a linear representation \rho whose character takes different values on g_1 and g_2.

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