turkmath.org

Given a finite group G, a celebrated theorem of Burnside which relates the influence of conjugacy class sizes of G on the group structure with irreducible characters, implies that any non-linear irreducible character of G vanishes on some element of G. Such an element is called a vanishing element of G and its associated conjugacy class is called a vanishing class of G. Indeed, a zero appears in the {i,j}-position of the character table of G whenever the i-th irreducible character, say X_i, vanishes on the j-th conjugacy class, say C_j. In this talk, we will evaluate the effects of these zeros on the group structure of G.