On Zeros of Characters of Finite Groups

Abstract

We survey some results concerning the distribution of zeros in the character table of a finite group and its influence on the structure of the group itself.

Groups with vanishing class size 𝑝

“…As mentioned in the Introduction, there exist groups G as in conclusion (b) of the above theorem (and of Theorem A) for which N Z(G)/Z(G) is nonabelian. Such a group is, for instance, the normalizer of a Sylow 2-subgroup in the Suzuki group Suz (8), which is a Frobenius group of order 2 6 · 7 whose vanishing classes have all size 2 6 . Moreover, considering now G = SL(2, 3), we have G = N H where N is a normal Sylow 2-subgroup and H a 2-complement, and G/Z(G) is a Frobenius group whose Frobenius kernel N/Z(G) has order 4; but the set of vanishing class sizes of G contains two numbers, 4 and 6, as in fact N does contain vanishing elements of G.…”

“…Isaacs, G. Navarro and T.R. Wolf in [13]: an element g of a finite group G is called a vanishing element of G if there exists an irreducible character of G taking the value 0 on g, and the conjugacy class of such an element is called a vanishing conjugacy class of G. Now, one can focus on a subset of cs(G) "filtered" by means of the irreducible characters of G, considering the set vcs(G) of vanishing conjugacy class sizes of G (see also [8] for a survey on results concerning vanishing elements and vanishing conjugacy classes). In view of the previous paragraph, a natural issue in this context can be to investigate the structure of finite groups having a unique (non-central) vanishing conjugacy class size.…”

On solvable groups with one vanishing class size

“…We mention, for instance, the studies by Itô [15] and Ishikawa [14] showing that a finite group with conjugate rank 1 is, up to abelian direct factors, a group of prime-power order of nilpotency class at most 3; also, the class of finite groups of conjugate rank 2 was studied by Dolfi and Jabara in [6]. Recently, this research area has been intertwined with character theory via the concept of vanishing elements, introduced by Isaacs et al [13]: an element g of a finite group G is called a vanishing element of G if there exists an irreducible character of G taking the value 0 on g, and the conjugacy class of such an element is called a vanishing conjugacy class of G. Now, one can focus on a subset of cs(G) 'filtered' by means of the irreducible characters of G, considering the set vcs(G) of vanishing conjugacy class sizes of G (see also [10] for a survey on results concerning vanishing elements and vanishing conjugacy classes). In view of the previous paragraph, a natural issue in this context can be to investigate the structure of finite groups having a unique (non-central) vanishing conjugacy class size.…”

Conjugacy classes and factorised groups

“…As a sample, groups with no vanishing p-elements were investigated by Dolfi, Pacifici, Sanus and Spiga in [23]; Moretó and Sangroniz classified in [32] groups whose irreducible characters vanish on "few" conjugacy classes. We refer the interested reader to the expository paper [22] for more information on the subject. It is to be said that the CFSG is usually needed in this development.…”

“…Several international research groups, among them the one of the supervisor Felipe, have deeply discussed this issue. The exhaustive report [15] due to Camina and Camina describes a general perspective about the subject until 2011, and the recent case of vanishing elements is treated in [22] by Dolfi, Pacifici and Sanus. In parallel with the previous developments, the research on groups which can be factorised as a product of subgroups has become increasingly relevant, in the universe of finite groups as well as in the infinite one. In this line, a good number of authors have carried out in-depth investigations with the purpose of understanding how some information about the subgroups that appear in the factorisation affects the whole group structure.…”