On the number of conjugacy classes of zeros of characters

On the number of zeros in the columns of the character table of a group

“…Qian's bound was improved in [9, Theorem A] as a consequence of results of [10]. In [9] we also conjectured that the derived length of a solvable group is bounded in terms of m(G). The main result in this paper shows that, using the classification of finite simple groups, much more can be said if we consider m * (G).…”

“…We also discuss some more analogs to results of [9]. For p-groups, there is a lower bound for the number of zeros in a row of the character table in terms of the degree of the corresponding character (see [9,Theorem C]).…”

“…In [9] we obtained group theoretical properties of G in terms of m(G), which was defined to be the maximum number of zeros in a row of the character table, and n(G), the minimum number of zeros in a row corresponding to a non-linear character of the character table. Motivated by this work, we study here the dual question.…”

Finite groups whose irreducible characters vanish on at most three conjugacy classes

“…Suppose that jGj d 16. As G is of maximal class, one of the upper central series members must have index 16. Now every group of order 16 has a non-linear irreducible character which vanishes on at least four conjugacy classes (see [12, p. 300]).…”

Degrees, class sizes and divisors of character values

“…Analogously, on each larger than average class the character values are roots of unity or zero for more than a third of the characters. Among recent results about many zeros in the character table, I. M. Isaacs, G. Navarro and T. R. Wolf [12] have shown that each noncentral element of a nilpotent group is a zero of some character, and A. Moreto and J. Sangroniz [19,20] have shown that there is a character of G vanishing on many classes if the Fitting height of G is large, and there is a class on which many characters vanish if the number of nonlinear characters is large. (ii) For each element x of G in a class at least as large as the average class size (i.e., with jC G .x/j Ä jCl.G/j), at least three quarters of the irreducible characters of G satisfy either d. / divides s.x/ or .x/ D 0.…”