Characters of π′-degree and small cyclotomic fields

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 Q ( e 2 π i / q ) . We further characterize when such character can be chosen with only rational values in solvable groups. These res...

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Bibliographic Details
Published in:Annali di matematica pura ed applicata 2021-06, Vol.200 (3), p.1055-1073
Main Authors: Giannelli, Eugenio, Hung, Nguyen Ngoc, Schaeffer Fry, A. A., Vallejo, Carolina
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
Prime numbers
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Summary: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 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 } ′ -degree nontrivial irreducible character with values in Q ( e 2 π i / p ) or 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 } ′ -degree nontrivial irreducible character with rational values.
ISSN:0373-3114
1618-1891
DOI:10.1007/s10231-020-01025-x