On products of groups and indices not divisible by a given prime

[EN] Let the group G = AB be the product of subgroupsAandB, and letpbe a prime. We prove thatpdoes not divide the conjugacy class size (index) of eachp-regular element of prime power order x is an element of A boolean OR B if and only if G is p-decomposable, i.e. G = Op(G) x Op '(G)Research supported by Proyecto PGC2018-096872-B-I00 from the Ministerio de Ciencia, Innovacion y Universidades, Spain, and FEDER. The second author is also supported by Project VIP-008 of Yaroslavl P. Demidov State University and the third author by Proyecto PROMETEO/2017/057 from the Generalitat Valenciana, SpainFelipe Román, MJ.; Kazarin, LS.; Martínez-Pastor, A.; Ortiz-Sotomayor, VM. (2020). On products of groups and indices not divisible by a given prime. Monatshefte für Mathematik. 193(4):811-827. https://doi.org/10.1007/s00605-020-01446-z8118271934Amberg, B., Franciosi, S., de Giovanni, F.: Products of Groups. Oxford University Press Inc., New York (1992)Ballester-Bolinches, A., Cossey, J., Li, Y.: Mutually permutable products and conjugacy classes. Monatsh. Math. 170, 305–310 (2013)Ballester-Bolinches, A., Esteban-Romero, R., Asaad, M.: Products of Finite Groups. Vol. 53 of de Gruyter Expositions in Mathematics. de Gruyter, Berlin (2010)Camina, A.R., Camina, R.D.: The influence of conjugacy class sizes on the structure of finite groups: a survey. Asian Eur. J. Math. 4, 559–588 (2011)Carter, R.W.: Centralizers of semisimple elements in the finite classical groups. Proc. Lond. Math. Soc. 42, 1–41 (1981)Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Clarendon Press, Oxford (1985)Cossey, J., Li, Y.: On the structure of a mutually permutable product of finite groups. Acta Math. Hungar. 154, 525–529 (2018)Doerk, K., Hawkes, T.: Finite Soluble Groups. Vol. 4 of de Gruyter Expositions in Mathematics. de Gruyter, Berlin (1992)Dolfi, S., Pacifici, E., Sanus, L., Spiga, P.: On the orders of zeros of irreducible characters. J. Algebra 321, 345–352 (2009)Fein, B., Kantor, W.M., Schacher, M.: Relative Brauer groups II. J. Reine Angew. Math. 328, 39–57 (1981)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.: Zeros of irreducible characters in factorised groups. Ann. Mat. Pura Appl. 198, 129–142 (2019)Felipe M.J., Martínez-Pastor A., Ortiz-Sotomayor V.M.: Products of groups and class sizes of $$\pi $$-elements. Mediterr. J. Math. 17, article: 15 (2020)He, H., Shi, W.: A note on the adjacency criterion for the prime graph and the characterization of $$C_p (3)$$. Algebra Colloq. 19(3), 553–562 (2012)Kazarin, L.S., Martínez-Pastor, A., Pérez-Ramos, M.D.: A reduction theorem for a conjecture on products of two $$\pi $$-decomposable groups. J. Algebra 379, 301–313 (2013)Kondrat’iev A.S.: On prime graph components of finite simple groups, Math. Sb. 67(1), 235–247 (1990) Translation from Mat. Sb. 180(6), 787–797 (1989)Liebeck, M., Praeger, C.E., Saxl, J.: The Maximal Factorizations of the Finite Simple Groups and Their Automorphism Groups, Mem. Amer. Math. Soc., 86 (432). Amer. Math. Soc, Providence (1990)Liu, X., Wang, Y., Wei, H.: Notes on the length of conjugacy classes of finite groups. J. Pure Appl. Algebra 196, 111–117 (2005)Lucido, M.S.: Prime graph components of finite almost simple groups. R. Semin. Mat. Univ. Padova 102, 1–22 (1999) Addendum in: R. Semin. Mat. Univ. Padova 107, 1–2 (2002)Vasiliev, A.V., Vdovin, E.P.: An adjacency criterion for the prime graph of a finite simple group. Algebra Log. 44(6), 381–406 (2005)Vasiliev, A.V., Vdovin, E.P.: Cocliques of maximal size in the prime graph of a finite simple group. Algebra Log. 50(4), 291–322 (2011)Williams, J.S.: Prime graph components of finite groups. J. Algebra 69, 487–513 (1981)Zavarnitsin, A.V.: Recognition of the simple groups $$L_3 (q)$$ by element orders. J. Group Theory 7, 81–97 (2004)Zhao, X.H., Guo, X.Y., Shi, J.Y.: On the conjugacy class sizes of prime power order $$\pi $$-elements. South. Asian Bull. Math. 35, 735–740 (2011)Zsigmondy, K.: Zur Theorie der Potenzreste. Monatsh. Math. Phys. 3, 265–284 (1892