Sharp Characters Whose Values at Non-identity Elements Are 0 and a Family of Algebraic Conjugates

AbstractCameron and Kiyota [J. Algebra115 (1988), 125-143] posed the problem of determining all the L-sharp pairs (G, χ) for a given set L, and they conjectured that G is dihedral at twice odd prime order if L is the set {0} ∪ L′, where L′ is a family of algebraic conjugates. In their paper and that of S. Nozawa [Tsukuba J. Math.16 (1992), 269-277] this conjecture is proved to be true under one of the following conditions: (1) χ(1) is coprime to ƒL′(n); (2) |L′| = 2; (3) χ is irreducible. We can now show that this conjecture is true without any additional conditions