On zeros of irreducible characters lying in a normal subgroup
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On zeros of irreducible characters lying in a normal subgroup M. J. Felipe1 · N. Grittini2 · V. Sotomayor1 Received: 28 July 2019 / Accepted: 11 January 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Let N be a normal subgroup of a finite group G. In this paper, we consider the elements g of N such that 𝜒(g) ≠ 0 for all irreducible characters 𝜒 of G. Such an element is said to be non-vanishing in G. Let p be a prime. If all p-elements of N satisfy the previous property, then we prove that N has a normal Sylow p-subgroup. As a consequence, we also study certain arithmetical properties of the G-conjugacy class sizes of the elements of N which are zeros of some irreducible character of G. In particular, if N = G , then new contributions are obtained. Keywords Finite groups · Normal subgroups · Irreducible characters · Conjugacy classes Mathematics Subject Classification 20C15 · 20E45
1 Introduction In the sequel, all groups considered are finite. Within character theory, a classical theorem of Burnside asserts that a nonlinear irreducible character of a finite group always vanishes on some element. It is not difficult to see that the converse is also true, so the rows of the character table of a group that contain a zero entry are completely characterised. However, the “dual” situation for conjugacy classes fails in general: a column that corresponds to a non-central conjugacy class may not contain a zero. This fact somehow violates the standard duality that in many cases arises between irreducible characters and conjugacy classes
* V. Sotomayor [email protected] M. J. Felipe [email protected] N. Grittini [email protected] 1
Instituto Universitario de Matemática Pura y Aplicada (IUMPA), Universitat Politècnica de València, Camino de Vera, s/n, 46022 Valencia, Spain
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Dipartimento di Matematica U. Dini, Università degli Studi di Firenze, Viale Morgagni 67/a, 50134 Florence, Italy
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of a group. Therefore, for a group G, an element g is said to be non-vanishing in G if 𝜒(g) ≠ 0 for every irreducible character 𝜒 of G. An immediate corollary to the aforementioned Burnside’s result is that a group is abelian if and only if every element is non-vanishing. Isaacs et al. [10] obtained elegant results about the location of non-vanishing elements in certain groups. For example, for a nilpotent group G, an element is non-vanishing if and only if it lies in the centre of G. They F(G) is a 2-element for a non-vanishing element g also proved that if G is soluble, then gF of G. Consequently, if g is of odd order, then x lies in F (G) . These authors conjectured that every non-vanishing element of a soluble group G lies in F (G) , and it is still an open problem. In this paper, we prove the following result which provides further evidence for this conjecture.
Theorem A Let N be a normal subgroup of a group G, and let p be a prime. If 𝜒(x) ≠ 0 for every p-element x ∈ N and for all 𝜒
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