Zeros of irreducible characters in factorised groups

[EN] An element g of a finite group G is said to be vanishing inG if there exists an irreducible character of G such that (g)=0; in this case, g is also called a zero of G. The aim of this paper is to obtain structural properties of a factorised group G=AB when we impose some conditions on prime power order elements gAB which are (non-)vanishing in G.The first author is supported by Proyecto PrometeoII/2015/011, Generalitat Valenciana (Spain). The second author is supported by Proyecto MTM2014-54707-C3-1-P, Ministerio de Economia, Industria y Competitividad (Spain), and by Proyecto Prometeo/2017/057, Generalitat Valenciana (Spain). The third author acknowledges the predoctoral grant ACIF/2016/170, Generalitat Valenciana Spain).Ortiz-Sotomayor, VM.; Felipe Román, MJ.; Martínez-Pastor, A. (2019). Zeros of irreducible characters in factorised groups. Annali di Matematica Pura ed Applicata (1923 -). 198:129-142. https://doi.org/10.1007/s10231-018-0765-5129142198Arroyo-Jordá, M., Arroyo-Jordá, P., Martínez-Pastor, A., Pérez-Ramos, M.D.: On conditional permutability and factorized groups. Ann. Mat. Pura Appl. 193, 1123–1138 (2014)Arroyo-Jordá, M., Arroyo-Jordá, P., Pérez-Ramos, M.D.: On conditional permutability and saturated formations. Proc. Edinb. Math. Soc. 54, 309–319 (2011)Ballester-Bolinches, A., Cossey, J., Li, Y.: Mutually permutable products and conjugacy classes. Monatsh. Math. 170, 305–310 (2012)Ballester-Bolinches, A., Esteban-Romero, R., Asaad, M.: Products of finite groups. Vol. 53 of de Gruyter Expositions in Mathematics, Berlin (2010)Bianchi, M., Brough, J., Camina, R.D., Pacifici, E.: On vanishing class sizes in finite groups. J. Algebra 489, 446–459 (2017)Brough, J.: Non-vanishing elements in finite groups. J. Algebra 460, 387–391 (2016)Brough, J.: On vanishing criteria that control finite group structure. J. Algebra 458, 207–215 (2016)Brough, J., Kong, Q.: On vanishing criteria that control finite group structure II. ArXiv (2017). arXiv:1702.03179Camina, A.R., Camina, R.D.: Implications of conjugacy class size. J. Group Theory 1(3), 257–269 (1998)Cossey, J., Li, Y.: On the structure of a mutually permutable product of finite groups. Acta Math. Hungar. 154(2), 525–529 (2018)Cossey, J., Wang, Y.: Remarks on the length of conjugacy classes of finite groups. Commun. Algebra 27, 4347–4353 (1999)Doerk, K., Hawkes, T.: Finite Soluble Groups. Vol. 4 of de Gruyter Expositions in Mathematics, Berlin (1992)Dolfi, S., Pacifici, E., Sanus, L.: Groups whose vanishing class sizes are not divisible by a given prime. Arch. Math. (Basel) 94, 311–317 (2010)Dolfi, S., Pacifici, E., Sanus, L., Spiga, P.: On the orders of zeros of irreducible characters. J. Algebra 321, 345–352 (2009)Felipe, M.J., Martínez-Pastor, A., Ortiz-Sotomayor, V.M.: Square-free class sizes in products of groups. J. Algebra 491, 190–206 (2017)Felipe, M.J., Martínez-Pastor, A., Ortiz-Sotomayor, V.M.: Prime power indices in factorised groups. Mediterr. J. Math. 14(6), article: 225 (2017)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., Olsson, J.B.: Zeros of characters of finite groups. J. Group Theory 3, 353–368 (2000