On zeros of irreducible characters lying in a normal subgroup

[EN] Let N be a normal subgroup of a finite group G. In this paper, we consider the elements g of N such that x(g)¿0 for all irreducible characters x 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 h...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Hauptverfasser: Felipe Román, María Josefa, Grittini, N, Sotomayor, Víctor
Format: Artikel
Sprache:eng
Schlagworte:
Online-Zugang:Volltext bestellen
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Beschreibung
Zusammenfassung:[EN] Let N be a normal subgroup of a finite group G. In this paper, we consider the elements g of N such that x(g)¿0 for all irreducible characters x 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. The first author is supported by Proyecto Prometeo II/2015/011, Generalitat Valenciana (Spain). The research of the second author is partially funded by the Istituto Nazionale di Alta Matematica - INdAM. The third author acknowledges the predoctoral grant ACIF/2016/170, Generalitat Valenciana (Spain). The first and third authors are also supported by Proyecto PGC2018-096872-B-I00, Ministerio de Ciencia, Innovacion y Universidades (Spain). Felipe Román, MJ.; Grittini, N.; Sotomayor, V. (2020). On zeros of irreducible characters lying in a normal subgroup. Annali di Matematica Pura ed Applicata (1923 -). 199:1777-1789. https://doi.org/10.1007/s10231-020-00942-1 Beltrán, A., Felipe, M.J.: Prime powers as conjugacy class lengths of $$\pi$$-elements. Bull. Aust. Math. Soc. 69, 317–325 (2004) Beltrán, A., Felipe, M.J., Malle, G., Moretó, A., Navarro, G., Sanus, L., Solomon, R., Tiep, P.H.: Nilpotent and abelian Hall subgroups in finite groups. Trans. Am. Math. Soc. 368, 2497–2513 (2016) Berkovich, Y., Kazarin, L.S.: Indices of elements and normal structure of finite groups. J. Algebra 283, 564–583 (2005) Bianchi, M., Chillag, D., Lewis, M.L., Pacifici, E.: Character degree graphs that are complete graphs. Proc. Am. Math. Soc. 135, 671–676 (2007) Brough, J., Kong, Q.: On vanishing criteria that control finite group structure II. Bull. Aust. Math. Soc. 98, 251–257 (2018) Brough, J.: Non-vanishing elements in finite groups. J. Algebra 460, 387–391 (2016) Dolfi, S., Pacifici, E., Sanus, L., Spiga, P.: On the orders of zeros of irreducible characters. J. Algebra 321, 345–352 (2009) Grüninger, M.: Two remarks about non-vanishing elements in finite groups. J. Algebra 460, 366–369 (2016) Isaacs, I.M.: Character Theory of Finite Groups. Academic Press Inc., London (1976) Isaacs, I.M., Navarro, G., Wolf, T.R.: Finite group elements where no irreducible character vanishes. J. Algebra 222, 413–423 (1999) Malle, G., Navarro, G.