Characters of $$\pi '$$-degree and small cyclotomic fields

Abstract

AbstractWe show that every finite group of order divisible by 2 or q, where q is a prime number, admits a $$\{2, q\}'$$ { 2 , q } ′ -degree nontrivial irreducible character with values in $${\mathbb{Q}}(e^{2 \pi i /q})$$ Q ( e 2 π 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 $$\{p, q\}'$$ { p , q } ′ -degree nontrivial irreducible character with values in $${\mathbb{Q}}(e^{2 \pi i /p})$$ Q ( e 2 π i / p ) or $${\mathbb{Q}}(e^{ 2 \pi i/q})$$ Q ( e 2 π i / q ) , for any pair of primes p and q. Along the way, we completely describe simple alternating groups admitting a $$\{p, q\}'$$ { p , q } ′ -degree nontrivial irreducible character with rational values.