Faithful generalized characters and generalized classes of a finite group
It turns out that the values of certain generalized characters, which we shall refer to in this paper as ``faithful'', by analogy with the characters of faithful representations, almost determine the group, in the sense that, as shown in Theorem 1, there exist only finitely many finite groups having faithful generalized characters with a given set of distinct values. The analogous question is considered for rational integer combinations of Brauer characters corresponding to a given prime \(p\) (Theorem 2). In both cases we obtain explicit bounds for the order of a group in terms of the values of its faithful generalized characters. Similar problems are solved for the generalized classes of a finite group, i.e., linear combinations of conjugate classes with rational integer coefficients. In particular, it will be shown that a group is almost determined by the spectrum of the operator defined on the center of the group algebra as multiplication by a faithful generalized class (Theorem 5).
scientific article; zbMATH DE number 4398