Abstract
The aim of this paper is to prove the following result: Let \(\pi \) be a set of odd primes. If the group \(G=AB\) is the product of two \(\pi \)-decomposable subgroups \(A=A_\pi \times A_{\pi '}\) and \(B=B_\pi \times B_{\pi '}\), then G has a unique conjugacy class of Hall \(\pi \)-subgroups, and any \(\pi \)-subgroup is contained in a Hall \(\pi \)-subgroup (i.e. G satisfies property \(D_{\pi }\)).
Similar content being viewed by others
References
Amberg, B., Franciosi, S., de Giovanni, F.: Products of Groups. Clarendon Press, Oxford (1992)
Amberg, B., Carocca, A., Kazarin, L.S.: Criteria for the solubility and non-simplicity of finite groups. J. Algebra 28(5), 58–72 (2005)
Arad, Z., Chillag, D.: Finite groups containing a nilpotent Hall subgroup of even order. Houston J. Math. 7, 23–32 (1981)
Arad, Z., Fisman, E.: On finite factorizable groups. J. Algebra 86, 522–548 (1984)
Berkovich, Y.G.: Generalization of the theorems of Carter and Wielandt. Sov. Math. Dokl. 7, 1525–1529 (1966)
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: Maximal Subgroups and Ordinary Characters of Simple Groups. Clarendon Press, Oxford (1985)
Doerk, K., Hawkes, T.: Finite Soluble Groups. Walter De Gruyter, Berlin, New York (1992)
Gross, F.: Odd order Hall subgroups of \(GL_n(q)\) and \(Sp_{2n}(q)\). Math. Z. 187, 185–194 (1984)
Gross, F.: Conjugacy of odd order Hall subgroups. Bull. Lond. Math. Soc. 19, 311–319 (1987)
Hall, P.: Theorems like Sylow’s. Proc. Lond. Math. Soc. 3(2), 286–304 (1956)
Huppert, B.: Endliche Gruppen I. Springer, Heidelberg, New York (1967)
Kazarin, L.S.: Criteria for the nonsimplicity of factorable groups. Izv. Akad. Nauk SSSR Ser. Mat. 44, 288–308 (1980)
Kazarin, L.S.: On groups which are the product of two soluble groups. Comm. Algebra 14, 1001–1066 (1986)
Kazarin, L.S.: On a problem of Szép. Math. USSR Izvestiya 28, 467–495 (1987)
Kazarin, L.S.: Factorizations of finite groups by solvable subgroups. Ukr. Mat. J. 43(7), 883–886 (1991)
Kazarin, L.S., Martínez-Pastor, A., Pérez-Ramos, M.D.: On the product of a \(\pi \)-group and a \(\pi \)-decomposable group. J. Algebra 315, 640–653 (2007)
Kazarin, L.S., Martínez-Pastor, A., Pérez-Ramos, M.D.: On the product of two \(\pi \)-decomposable soluble groups. Publ. Mat. 53, 439–456 (2009)
Kazarin, L.S., Martínez-Pastor, A. Pérez-Ramos, M.D.: Extending the Kegel-Wielandt theorem through \(\pi \)-decomposable groups. In: Groups St Andrews 2009 in Bath, vol. 2, Lond. Math. Soc. Lecture Note Ser. 388, pp. 415–423, Cambridge University Press, Cambridge (2011)
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)
Kazarin, L.S., Martínez-Pastor, A., Pérez-Ramos, M.D.: On the product of two \(\pi \)-decomposable groups. Rev. Mat. Iberoam. 31, 51–68 (2015)
Kazarin, L.S., Martínez-Pastor, A., Pérez-Ramos, M.D.: Finite trifactorized groups and \(\pi \)-decomposability. Bull. Aust. Math. Soc. 97, 218–228 (2018)
Kleidman, P., Liebeck, M.: The Subgroup Structure of the Finite Classical Groups. Cambridge University Press, Cambridge (1990)
Li, C.H., Xia, B.: Factorizations of almost simple groups with a solvable factor, and Cayley graphs of solvable groups. To appear in Mem. Am. Math. Soc. arXiv:1408.0350
Liebeck, M., Praeger, C.E., Saxl, J.: The maximal factorizations of the finite simple groups and their automorphism groups. Mem. Am. Math. Soc. 86, 432 Am. Math. Soc., Providence, RI (1990)
Revin, D.O., Vdovin, E.P.: Hall subgroups in finite groups. In: Ischia Group Theory 2004, Contemp. Math. 402, pp. 229–263. Am. Math. Soc., Providence, RI (2006)
Rowley, P.J.: The \(\pi \)-separability of certain factorizable groups. Math. Z. 153, 219–228 (1977)
Vasiliev, A.V., Vdovin, E.P.: An adjacency criterion for the prime graph of a finite simple group. Algebra Logic 44(6), 381–406 (2005)
Vdovin, E.P., Revin, D.O.: Theorems of Sylow type. Russ. Math. Surv. 66(5), 829–870 (2011)
Wielandt, H.: Zum Satz von Sylow. Math. Z. 60, 407–408 (1954)
Zsigmondy, K.: Zur Theorie der Potenzreste. Monatsh. Math. Phys. 3, 265–284 (1892)
Acknowledgements
We thank the reviewers for their helpful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Research supported by Proyectos PROMETEO/2017/057 from the Generalitat Valenciana (Valencian Community, Spain), and PGC2018-096872-B-I00 from the Ministerio de Ciencia, Innovación y Universidades, Spain, and FEDER, European Union; and second author also by Project VIP-008 of Yaroslavl P. Demidov State University.
Rights and permissions
About this article
Cite this article
Kazarin, L.S., Martínez-Pastor, A. & Pérez-Ramos, M.D. The \(D_{\pi }\)-property on products of \(\pi \)-decomposable groups. RACSAM 115, 13 (2021). https://doi.org/10.1007/s13398-020-00950-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13398-020-00950-z