Abstract:
Fix a set of primes π. A finite group is said to satisfy Cπ or, in other words, to be a Cπ-group, if it possesses exactly one class of conjugate π-Hall subgroups. The pronormality of π-Hall subgroups in Cπ-groups is proven, or, equivalently, we show that Cπ is inherited by overgroups of π-Hall subgroups. Thus an affirmative solution is obtained to Problem 17.44(a) from The Kourovka Notebook. We also provide some example demonstrating that Hall subgroups in finite groups are not pronormal in general.
Keywords:
pronormal subgroup, π-Hall subgroup, Hall properties Eπ, Cπ, and Dπ.
Citation:
E. P. Vdovin, D. O. Revin, “On the pronormality of Hall subgroups”, Sibirsk. Mat. Zh., 54:1 (2013), 35–43; Siberian Math. J., 54:1 (2013), 22–28
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\paper On the pronormality of Hall subgroups
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\pages 35--43
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\jour Siberian Math. J.
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Linking options:
https://www.mathnet.ru/eng/smj2397
https://www.mathnet.ru/eng/smj/v54/i1/p35
This publication is cited in the following 15 articles:
E. P. Vdovin, N. Ch. Manzaeva, D. O. Revin, “On the heritability of the Sylow $\pi$-theorem by subgroups”, Sb. Math., 211:3 (2020), 309–335
Guo Wen Bin, A. A. Buturlakin, D. O. Revin, “Equivalence of the existence of nonconjugate and nonisomorphic Hall $\pi$-subgroups”, Proc. Steklov Inst. Math. (Suppl.), 303, suppl. 1 (2018), 94–99
W. Guo, D. O. Revin, “Pronormality and submaximal $\mathfrak{X}$-subgroups on finite groups”, Commun. Math. Stat., 6:3, SI (2018), 289–317
W. Guo, D. O. Revin, “Classification and properties of the $\pi$-submaximal subgroups in minimal nonsolvable groups”, Bull. Math. Sci., 8:2 (2018), 325–351
E. P. Vdovin, M. N. Nesterov, D. O. Revin, “Pronormality of Hall subgroups in their normal closure”, Algebra and Logic, 56:6 (2018), 451–457
M. N. Nesterov, “On pronormality and strong pronormality of Hall subgroups”, Siberian Math. J., 58:1 (2017), 128–133
E. P. Vdovin, D. O. Revin, “Abnormality criteria for $p$-complements”, Algebra and Logic, 55:5 (2016), 347–353
E. P. Vdovin, D. O. Revin, “The existence of pronormal $\pi$-Hall subgroups in $E_\pi$-groups”, Siberian Math. J., 56:3 (2015), 379–383
A. S. Kondrat'ev, N. V. Maslova, D. O. Revin, “On the pronormality of subgroups of odd index in finite simple groups”, Siberian Math. J., 56:6 (2015), 1101–1107
M. N. Nesterov, “Pronormalnost khollovykh podgrupp v pochti prostykh gruppakh”, Sib. elektron. matem. izv., 12 (2015), 1032–1038
N. Ch. Manzaeva, “Heritability of the property $\mathcal D_\pi$ by overgroups of $\pi$-Hall subgroups in the case where $2\in\pi$”, Algebra and Logic, 53:1 (2014), 17–28
W. Guo, D. O. Revin, “On the class of groups with pronormal Hall $\pi$-subgroups”, Siberian Math. J., 55:3 (2014), 415–427
Revin D.O. Vdovin E.P., “Frattini Argument For Hall Subgroups”, J. Algebra, 414 (2014), 95–104
E. P. Vdovin, D. O. Revin, “Pronormality and strong pronormality of subgroups”, Algebra and Logic, 52:1 (2013), 15–23
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