Geodesic
Conjugacy of maximal and submaximal $\mathfrak X$-subgroups
Algebra i logika, Tome 57 (2018) no. 3, pp. 261-278.

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Let $\mathfrak X$ be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. Following H. Wielandt, we call a subgroup $H$ of a finite group $G$ a submaximal $\mathfrak X$-subgroup if there exists an isomorpic embedding $\phi\colon G\hookrightarrow G^*$ of the group $G$ into some finite group $G^*$ under which $G^\phi$ is subnormal in $G^*$ and $H^\phi=K\cap G^\phi$ for some maximal $\mathfrak X$-subgroup $K$ of $G^*$. We discuss the following question formulated by Wielandt: Is it always the case that all submaximal $\mathfrak X$-subgroups are conjugate in a finite group $G$ in which all maximal $\mathfrak X$-subgroups are conjugate? This question strengthens Wielandt's known problem of closedness for the class of $\mathscr D_\pi$-groups under extensions, which was solved some time ago. We prove that it is sufficient to answer the question mentioned for the case where $G$ is a simple group.
Keywords: finite group, maximal $\mathfrak X$-subgroup, submaximal $\mathfrak X$-subgroup, Hall $\pi$-subgroup, $\mathscr D_\pi$-property, $\mathscr D_\mathfrak X$-property. 
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W. Guo; D. O. Revin. Conjugacy of maximal and submaximal $\mathfrak X$-subgroups. Algebra i logika, Tome 57 (2018) no. 3, pp. 261-278. http://geodesic.mathdoc.fr/item/AL_2018_57_3_a0/

[1] V. Go, D. O. Revin, “O maksimalnykh i submaksimalnykh $\mathfrak X$-podgruppakh”, Algebra i logika, 57:1 (2018), 14–42 | MR | Zbl

[2] H. Wielandt, “Zusammengesetzte Gruppen: Holders Programm heute”, Finite groups (Santa Cruz Conf., 1979), Proc. Symp. Pure Math., 37, Am. Math. Soc., Providence, RI, 1980, 161–173 | DOI | MR | Zbl

[3] H. Wielandt, “Zusammengesetzte Gruppen endlicher Ordnung. Vorlesung an der Univ. Tübingen im Wintersemester 1963/64”: H. Wielandt, Mathematische Werke. Mathematical works, v. 1, Group theory, eds. B. Huppert, H. Schneider, Walter de Gruyter, Berlin, 1994, 607–655 | MR

[4] D. O. Revin, E. P. Vdovin, “Hall subgroups of finite groups”, Ischia group theory 2004, Proc. conf. honor Marcel Herzog (Naples, Italy, March 31 – April 03, 2004), Contemp. Math., 402, Israel Math. Conf. Proc., eds. Arad Zvi et al., Am. Math. Soc., Providence, RI; Bar-Ilan Univ., Ramat Gan, 2006, 229–263 | DOI | MR | Zbl

[5] E. P. Vdovin, D. O. Revin, “Teoremy silovskogo tipa”, Uspekhi matem. n., 66:5(401) (2011), 3–46 | DOI | MR | Zbl

[6] W. Guo, Structure theory of canonical classes of finite groups, Springer-Verlag, Berlin, 2015 | MR | Zbl

[7] H. Wielandt, “Entwicklungslinien in der Strukturtheorie der endlichen Gruppen,”, Proc. Int. Congr. Math. (Edinburgh, 14–21 Aug. 1958), Cambridge Univ. Press, New York, 1960, 268–278 | MR

[8] V. D. Mazurov, D. O. Revin, “O khollovom $D_\pi$-svoistve dlya konechnykh grupp”, Sib. matem. zh., 38:1 (1997), 125–134 | MR | Zbl

[9] W. Feit, J. G. Thompson, “Solvability of groups of odd order”, Pac. J. Math., 13:3 (1963), 775–1029 | DOI | MR | Zbl

[10] H. Wielandt, “Eine Verallgemeinerung der invarianten Untergruppen”, Math. Z., 45 (1939), 209–244 | DOI | MR | Zbl

[11] P. Hall, “Theorems like Sylow's”, Proc. Lond. Math. Soc. III Ser., 6 (1956), 286–304 | DOI | MR | Zbl

[12] D. O. Revin, E. P. Vdovin, “On the number of classes of conjugate Hall subgroups in finite simple groups”, J. Algebra, 324:12 (2010), 3614–3652 | DOI | MR | Zbl