Revisiting Character Theory of Finite Groups

Abstract

Two conjectures proposed (old and somewhat new) by the author elsewhere are discussed in this article. One is concerned with a modular version of the regular character of a finite group G, and the second one is concerned with the ratio of the product of the sizes of all conjugacy classes of G and the product of the degrees of all irreducible characters. 1. Conjectures I and II Let G be a finite group and Irr(G) = {χ1, χ2, . . . , χs} be the set of all inequivalent irreducible characters of G. Furthermore, let Conj(G) = {K1,K2, . . . ,Ks} be the set of all conjugacy classes of G. Choose a representative xj ∈ Kj for each j = 1, . . . , s and choose once and for all, χ1 = 1 and K1 = {1}. The group G acts on the set G by (left) multiplication ρ(g) : G x → gx ∈ G. The corresponding (permutation) character ρG is called the regular character of G and it satisfies ρG = s ∑