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(e2πi/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(e2πi/p) or Q(e2πi/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.
Characters of π’-degree and small cyclotomic fields / eugenio giannelli, n'goc hung, mandi schaeffer fry, carolina vallejo. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 1618-1891. - STAMPA. - 200:(2021), pp. 1055-1073. [10.1007/s10231-020-01025-x]
Characters of π’-degree and small cyclotomic fields
eugenio giannelli;carolina vallejo
2021
Abstract
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(e2πi/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(e2πi/p) or Q(e2πi/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.File | Dimensione | Formato | |
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