Induced characters which are multiples of irreducibles

Primitive Characters and Permutation Characters of Solvable Groups

“…Berger's theorem for groups of odd order is the case e = 1. After the original version of this paper was written, it was kindly pointed out to us by I. M. Isaacs that this result is essentially the odd order case of a theorem of Ferguson and Isaacs, Theorem A in [8].…”

“…Using our results, we give a nice proof of a theorem of Isaacs on the restriction of certain quasi-primitive characters to normal subgroups, in the case |G| odd (see Corollary I), and we show that an analogous result holds in some cases for nonnormal subgroups (Theorem J). We also give a proof, in the case that G has odd order, of a result of Ferguson and Isaacs (their result is Theorem A in [8]), showing that multiples of primitive characters also cannot be induced from proper subgroups (see Theorem K) 1 . In Corollaries L -Q, we give various properties of characters of solvable groups which are direct consequences of the existence of an expression ( †).…”

Virtual characters on L-functions

“…So could this be one approach towards making in roads to Artin's conjecture ? Note that for the case of solvable groups, by the result of Ferguson and Isaacs [8] we can conclude that, if G is solvable and any positive integer multiple of an irreducible character χ is monomial, then χ itself is monomial. However for monomial characters we know Artin's holomorphy conjecture to hold, from the day Artin L-functions were discovered in 1923.…”

Heilbronn characters

“…It follows immediately then that kχ is monomial for some positive integer k. By Brauer's Theorem every L-function is meromorphic, so if some positive integer power of it is analytic, then the L-function itself is analytic. By a result of Ferguson and Isaacs, [FI89], if G is solvable and any positive integer multiple of an irreducible character χ is monomial, then χ itself is monomial.…”