The 2-Characters of a Group and the Weak Cayley Table

Group character theory has many applications, but it is not easy to characterize the information contained in the character table of a group, or give concise information which in addition to that in the character table determines a group. The first part of this chapter examines the extra information which is obtained if the irreducible 2-characters of a group are known. It is shown that this is equivalent to the knowledge of the “weak Cayley table” (WCT) of the group G. A list of properties which are determined by the WCT and not by the character table is obtained, but non-isomorphic groups may have the same WCT. This work gives insight into why the problems in R. Brauer, Representations of finite groups, in “Lectures in Modern Mathematics, Vol. I,” T.L. Saaty (ed.), Wiley, New York, NY, 1963, 133–175, have often been hard to answer. There is a discussion of the work of Humphries on W(G) the group of weak Cayley table isomorphisms (WCTI’s). This group contains the a subgroup W₀(G) generated by Aut(G) and the anti-automorphism g → g⁻¹. W(G) is said to be trivial if W(G) = W₀(G). For some groups, such as the symmetric groups and dihedral groups W(G), is trivial, but there are many examples where W(G) is much larger than W₀(G). The discussion is not limited to the finite case. The category of “weak morphisms” between groups is described. These are weaker than ordinary homomorphisms but preserve more structure than maps which preserve the character table. A crossed product condition on these morphisms is needed to make this category associative. This category has some interesting properties.