WEAK CAYLEY TABLES
In  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) [rightward arrow] [open face C] for k = 1, …, degχ with χ(1) = χ. These functions are called the k-characters (see  or  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 . Summaries of this work are given in  and .(Received March 21 1997)
(Revised January 22 1999)
p1 Current address: Dipartimento di Matematica, Università di Trento, I-38050 POVO (Trento), Italy; firstname.lastname@example.org
1 The first author was partially supported by the Mathematical Research Institute at the Ohio State University.