It is known that for any set π of prime numbers, the following assertions are equivalent: (1) in any finite group, π-Hall subgroups are conjugate; (2) in any finite group, π-Hall subgroups are pronormal. It is proved that (1) and (2) are equivalent also to the following: (3) in any finite group, π-Hall subgroups are pronormal in their normal closure. Previously [10, Quest. 18.32], the question was posed whether it is true that in a finite group, π-Hall subgroups are always pronormal in their normal closure. Recently, M. N. Nesterov [7] proved that assertion (3) and assertions (1) and (2) are equivalent for any finite set π. The fact that there exist examples of finite sets π and finite groups G such that G contains more than one conjugacy class of π-Hall subgroups gives a negative answer to the question mentioned. Our main result shows that the requirement of finiteness for π is unessential for (1), (2), and (3) to be equivalent.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. P. Vdovin and D. O. Revin, “Theorems of Sylow type,” Usp. Mat. Nauk, 66, No. 5 (401), 3-46 (2011).
E. P. Vdovin and D. O. Revin, “The existence of pronormal π-Hall subgroups in E π -groups,” Sib. Math. J., 56, No. 3, 379-383 (2015).
E. P. Vdovin and D. O. Revin, “On the pronormality of Hall subgroups,” Sib. Math. J., 54, No. 1, 22-28 (2013).
W. Guo and D. O. Revin, “On the class of groups with pronormal Hall π-subgroups,” Sib. Math. J., 55, No. 3, 415-427 (2014).
E. P. Vdovin and D. O. Revin, “Pronormality of Hall subgroups in finite simple groups,” Sib. Math. J., 53, No. 3, 419-430 (2012).
E. P. Vdovin and D. O. Revin, “Pronormality and strong pronormality of subgroups,” Algebra and Logic, 52, No. 1, 15-23 (2013).
M. N. Nesterov, “Pronormality of Hall subgroups in almost simple groups,” Sib. El. Math. Izv., 12, 1032-1038 (2015); http://semr.math.nsc.ru/v12/p1032-1038.pdf.
M. N. Nesterov, “On pronormality and strong pronormality of Hall subgroups,” Sib. Math. J., 58, No. 1, 128-133 (2017).
E. P. Vdovin and D. O. Revin, “Abnormality criteria for p-complements,” Algebra and Logic, 55, No. 5, 347-353 (2016).
Unsolved Problems in Group Theory, The Kourovka Notebook, 18th edn., Institute of Mathematics SO RAN, Novosibirsk (2014); http://www.math.nsc.ru/∼alglog/18kt.pdf.
F. Gross, “Conjugacy of odd order Hall subgroups,” Bull. London Math. Soc., 19, No. 4, 311-319 (1987).
I. M. Isaacs, “Irreducible products of characters,” J. Alg., 223, No. 2, 630-646 (2000).
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford (1985).
M. Aschbacher, Finite Group Theory, Cambridge Univ. Press, Cambridge (1986).
M. D. Hestenes, “Singer groups,” Can. J. Math., 22, No. 3, 492-513 (1970).
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by RFBR, project No. 17-51-45025.
Translated from Algebra i Logika, Vol. 56, No. 6, pp. 682-690, November-December, 2017.
Rights and permissions
About this article
Cite this article
Vdovin, E.P., Nesterov, M.N. & Revin, D.O. Pronormality of Hall Subgroups in Their Normal Closure. Algebra Logic 56, 451–457 (2018). https://doi.org/10.1007/s10469-018-9467-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10469-018-9467-8