On sharp characters of type {−1, 0, 2}

Abstract

For a complex character χ of a finite group G, it is known that the product sh(χ)=∏l∈L(χ)(χ(1)−l) is a multiple of ∣G∣, where L(χ) is the image of χ on G − {1} The character χ is said to be a sharp character of type L if L = L(χ) and sh(χ) = ∣G∣. If the principal character of G is not an irreducible constituent of χ, then the character χ is called normalized. It is proposed as a problem by P. J. Cameron and M. Kiyota, to find finite groups G with normalized sharp characters of type {−1, 0, 2}. Here we prove that such a group with nontrivial center is isomorphic to the dihedral group of order 12. © 2022, Institute of Mathematics, Czech Academy of Sciences.