Number of irreducible representations over complex numbers with rational character values equals number of conjugacy classes of rational elements for any finite group whose cyclotomic splitting field is a cyclic extension of the rationals

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Statement

Suppose G is a finite group whose cyclotomic splitting field is a cyclic extension of the rationals. Then, the following numbers are equal:

  1. The number of irreducible representations of G over the complex numbers whose characters are rational-valued. Note that this includes both rational representations and representations realized over an extension that still have all their character values rational.
  2. The number of conjugacy classes in G of rational elements, i.e., elements that are conjugate to any other element that generates the same cyclic subgroup.

Related facts

Opposite facts

The result is not valid for all finite groups: Number of irreducible representations over complex numbers with rational character values need not equal number of conjugacy classes of rational elements

Similar facts

The following hold for any finite group: