Weak Cayley Tables

In 1] Brauer puts forward a series of questions on group representationtheory in order to point out areas which were not well understood.One of these, which we denote by (B1), is the following: whatinformation in addition to the character table determines a(finite) group? In previous papers 5, 7–13], the originalwork of Frobenius on group characters has been re-examined andhas shed light on some of Brauer's questions, in particularan answer to (B1) has been given as follows. Frobenius defined for each character of a group G functions^(k):G^(k) C for k = 1, ..., deg with ⁽¹⁾ = . These functionsare called the k-characters (see 10] or 11] for their definition).The 1-, 2- and 3-characters of the irreducible representationsdetermine a group 7, 8] but the 1- and 2-characters do not12]. Summaries of this work are given in 11] and 13].