NON-ISOMORPHIC GROUPS WITH ISOMORPHIC SPECTRAL TABLES AND BURNSIDE MATRICES

It was shown by Formanek and sibley that the group determinant characterizes a finite group G up to isomorphism. Hoehnke and Johnson (independently the authors - using an argument of Mansfield) showed the corresponding result for k-characters, k = 1, 2, 3. The notion of k-characters dates back to Frobenius. They are determined by the group determinant and may be derived from the character table CT(G) provided one knows additionally the functions PHI(k) : G x ... x G --> C(G), (g1, . . . , g(k)) --> Cg1.....g(k), where C(G) = {C(g), g is-an-element-of G} denotes the set of conjugacy classes of G. The object of the paper is to present criteria for finite groups (more precisely for soluble groups G and H which are both semi-direct products of a similar type) when 1. G and H have isomorphic spectral tables (i.e., they form a Brauer pair), 2. G and H have isomorphic table of marks (in particular the Burnside rings are isomorphic), 3. G and H have the same 2-characters. Using this the authors construct two non-isomorphic soluble groups for which all these three representation-theoretical invariants coincide.