In [1] Brauer puts forward a series of questions on group representation theory
in order to point out areas which were not well understood. One of these, which we
denote by (B1), is the following: what information in addition to the character table
determines a (finite) group? In previous papers [5, 7–13], the original work of
Frobenius on group characters has been re-examined and has shed light on some of
Brauer's questions, in particular an answer to (B1) has been given as follows.
Frobenius defined for each character χ of a group G functions
χ(k)[ratio ]G(k) → [Copf ] for
k = 1, …, degχ with χ(1) = χ. These functions are
called the k-characters (see [10] or
[11] for their definition). The 1-, 2- and 3-characters of the irreducible representations
determine a group [7, 8] but the 1- and 2-characters do not [12].
Summaries of this work are given in [11] and [13].