We show that every finite group of order divisible by 2 or q, where q is a prime number, admits a \(\{2, q\}'\)-degree nontrivial irreducible character with values in \({\mathbb{Q}}(e^{2 \pi i /q})\). We further characterize when such character can be chosen with only rational values in solvable groups. These results follow from more general considerations on groups admitting a \(\{p, q\}'\)-degree nontrivial irreducible character with values in \({\mathbb{Q}}(e^{2 \pi i /p})\) or \({\mathbb{Q}}(e^{ 2 \pi i/q})\), for any pair of primes p and q. Along the way, we completely describe simple alternating groups admitting a \(\{p, q\}'\)-degree nontrivial irreducible character with rational values.

我们表明，每个可被2或q整除的阶的有限组，其中q是质数，都接受一个\（\ {2，q \}'\）-级非平不可约字符，其值为\（{\ mathbb {Q }}（e ^ {2 \ pi i / q}）\）。我们进一步表征了何时只能以可解组的理性值来选择这种特征。这些结果来自对允许\（\ {p，q \}'\）度非平凡不可约字符的组具有\（{\ mathbb {Q}}（e ^ {2 \ pi i / p }）\）或\（{\ mathbb {Q}}（e ^ {2 \ pi i / q}）\），对于任意一对素数p和q。在此过程中，我们完整地描述了简单的交替组，它们允许具有合理值的\（\ {p，q \}'\）度非平凡不可约性。