Geodesic
On $p$-locally N-closed formations of finite groups
Trudy Instituta matematiki, Tome 18 (2010) no. 1, pp. 92-98.

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All groups considered are finite. A formation $\mathfrak{F}\ne\emptyset$ is called locally N-closed (N-closed) in a group class $\mathfrak{X}$, if the following assertion holds: if $G\in\mathfrak{X}$ and $P\le Z_{\mathfrak{F}}(N_G(P))$ ($N_G(P)\in \mathfrak{F}$ respectively) for every Sylow subgroup $P\ne1$ in $G$, then $G\in\mathfrak{F}$. It is proved that in the soluble universe, every hereditary saturated locally N-closed non-empty formation is N-closed. It is proved that the formation of all supersoluble groups is N-closed in the class of all soluble groups with $p$-length $\le1$ for every prime $p,$ and is not N-closed in the class of all soluble groups with $p$-length $\le2$ for every prime $p$. The authors also consider $p$-locally N-closed formations. 
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A. A. Rodionov; L. A. Shemetkov. On $p$-locally N-closed formations of finite groups. Trudy Instituta matematiki, Tome 18 (2010) no. 1, pp. 92-98. http://geodesic.mathdoc.fr/item/TIMB_2010_18_1_a9/

[1] Shemetkov L.A., Ballester-Bolinshe A., “O normalizatorakh silovskikh podgrupp v konechnykh gruppakh”, Sib. matem. zhurn., 40:1 (1999), 3–5 | MR | Zbl

[2] Bianchi M.G., Gillio Berta Mauri, Hauck P., “On finite groups with nilpotent Sylow normalizers”, Arch. Math., 47 (1986), 193–197 | DOI | MR | Zbl

[3] D'Aniello A., De Vivo C., Giordano G., “Finite groups with primitive Sylow normalizers”, Boll. U. M. I., 87 (2002), 3–13

[4] D'Aniello A., De Vivo C., Giordano G., “Saturated formations and Sylow normalizers”, Bull. Austral. Math. Soc., 69 (2004), 25–33 | DOI | MR

[5] D'Aniello A., De Vivo C., Giordano G., Pérez-Ramos M.D., “Saturated formations closed under Sylow normalizers”, Comm. Algebra, 33 (2005), 2801–2808 | DOI | MR

[6] Kazarin L.S., Martínez-Pastor A., Pérez-Ramos M.D., On the Sylow graph of a group and Sylow normalizers, 15 Dec 2009, 20 pp., arXiv: 0912.2839v1 [math. GR]

[7] Doerk K., Hawkes T.O., Finite Soluble Groups, Walter de Gruyter, Berlin, New York, 1992 | MR

[8] Shemetkov L.A., Formatsii konechnykh grupp, Nauka, M., 1978 | MR | Zbl

[9] Shemetkov L.A., Skiba A.N., Formatsii algebraicheskikh sistem, Nauka, M., 1989 | MR

[10] Semenchuk V.N., “Minimalnye ne $\mathfrak{F}$-gruppy”, Algebra i logika, 18:3 (1979), 348–382 | MR | Zbl

[11] Go Venbin, Shemetkov L.A., “O konechnykh gruppakh s gipertsentralnym usloviem”, Dokl. AN Belarusi, 36:6 (1992), 485–486 | MR

[12] Ballester-Bolinches A., Guo X., “Some results on $p$-nilpotence and solubility of finite groups”, J. Algebra, 228 (2000), 491–496 | DOI | MR | Zbl

[13] Shi W., “A note on $p$-nilpotence of finite groups”, J. Algebra, 241 (2001), 435–436 | DOI | MR | Zbl

[14] Shi J., Shi W., Zhang C., “A note on $p$-nilpotence and solvability of finite groups”, J. Algebra, 321 (2009), 1555–1560 | DOI | MR | Zbl