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A Generalization of the Aramata-Brauer Theorem
Sandra L. Rhoades
Proceedings of the American Mathematical Society
Vol. 119, No. 2 (Oct., 1993), pp. 357-364
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/2159915
Page Count: 8
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The Aramata-Brauer Theorem says that the regular character minus the principal character of a finite group can be written as a positive rational linear combination of induced linear characters. In the language of Artin L-series this says that ζE(s)/ζF(s) is entire, where this is the quotient of the Dedekind ζ-functions of a Galois extension E/F of number fields. Given any subset of characters of a finite group, we will give a necessary and sufficient condition for when a character is a positive rational linear combination of characters from this specified subset. This result implies that the regular character plus or minus any irreducible character can be written as a positive rational linear combination of induced linear characters. This both generalizes and gives a new proof of the Aramata-Brauer Theorem.
Proceedings of the American Mathematical Society © 1993 American Mathematical Society