Sharp characters of finite groups having prescribed values (Q1202807)

Let \(\chi\) be a virtual character of the finite group \(G\). Let \(L = \{\chi(g) \mid 1 \neq g \in G\}\) and let \(f_ L(x)\) be the polynomial \(f_ L(x) = \prod_{\ell \in L}(x - \ell)\). By a theorem of Blichfeldt we have \(| G|\mid f_ L(\chi(1))\), even more: \(f_ L(\chi(\cdot))\) is an integer multiple of the regular character of \(G\). The pair \((G,\chi)\) is called \(L\)-sharp if \(| G| = f_ L(\chi(1))\). Characterizing \(L\)-sharp pairs one may assume that the trivial character \(1_ G\) is not a constituent of \(\chi\) since adding \(1_ G\) to \(\chi\) does not affect the sharpness condition. In the paper under review the author proves the following two results: Theorem 1. Let \(\chi\) be a faithful character of the finite group \(G\) with \((\chi,1_ G) = 0\). Suppose that \((G,\chi)\) is \(L\)-sharp for \(L = \{\ell,\ell + 1,\ell + 2, \ell +3\}\) with \(\ell \in \mathbb{Z}\). Then (1) \(\ell = -2\) or \(-1\) and \(\chi\) is irreducible, (2) \(G \cong\text{SL}(2,3)\), \(S_ 5\), \(A_ 6\) or \(M_{11}\). Theorem 2. Replace \(L\) of Theorem 1 by \(L = \{0\} \cup L'\) where \(L'\) is a family of algebraic conjugates and \(| L'| \geq 2\). Then \(G\) is a dihedral group of order \(2p\) for a prime \(p\) and \(\chi\) is irreducible of degree 2.