Finite groups with all $n$-maximal ($n = 2, 3$) subgroups $K$-$\mathfrak{U}$-subnormal
Problemy fiziki, matematiki i tehniki, no. 2 (2014), pp. 59-63.

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A full classification of finite groups with all $n$-maximal ($n = 2, 3$) subgroups $K$-$\mathfrak{U}$-subnormal is given.
Keywords: $n$-maximal subgroup, $K$-$\mathfrak{U}$-subnormal subgroup, $\mathfrak{U}$-subnormal subgroup, supersoluble group, minimal nonsupersoluble group
Mots-clés : $SDH$-group.
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V. A. Kovaleva; Xiaolan Yi. Finite groups with all $n$-maximal ($n = 2, 3$) subgroups $K$-$\mathfrak{U}$-subnormal. Problemy fiziki, matematiki i tehniki, no. 2 (2014), pp. 59-63. http://geodesic.mathdoc.fr/item/PFMT_2014_2_a9/

[1] L. Rédei, “Ein Satz uber die endlichen einfachen Gruppen”, Acta Math., 84 (1950), 129–153 | DOI | MR | Zbl

[2] B. Huppert, “Normalteiler and maximal Untergruppen endlicher gruppen”, Math. Z., 60 (1954), 409–434 | DOI | MR | Zbl

[3] L. Ja. Poljakov, “Finite groups with permutable subgroups”, Finite groups, Proc. Gomel Sem., Nauka i Tekhnika, Minsk, 1966, 75–88 | MR

[4] R. K. Agrawal, “Generalized center and hypercenter of a finite group”, Proc. Amer. Math. Soc., 54 (1976), 13–21 | DOI | MR

[5] Z. Janko, “Finite groups with invariant fourth maximal subgroups”, Math. Zeitschr., 82 (1963), 82–89 | DOI | MR | Zbl

[6] M. Suzuki, “The nonexistence of a certain type of simple groups of odd order”, Proc. Amer. Math. Soc., 8:4 (1957), 686–695 | DOI | MR

[7] Z. Janko, “Endliche Gruppen mit lauter nilpotent zweitmaximalen Untergruppen”, Math. Z., 79 422–424 (1962), 79. - P. 422–424 | DOI | MR

[8] T. M. Gagen, Z. Janko, “Finite simple groups with nilpotent third maximal subgroups”, J. Austral. Math. Soc., 6:4 (1966), 466–469 | DOI | MR | Zbl

[9] V. A. Belonogov, “Finite soluble groups with nilpotent 2-maximal subgroups”, Math. Notes, 3:1 (1968), 15–21 | DOI | MR

[10] V. N. Semenchuk, “Soluble groups with supersoluble second maximal subgroup”, Voprosy Algebry, 1 (1985), 86–96 | MR | Zbl

[11] A. Mann, “Finite groups whose $n$-maximal subgroups are subnormal”, Trans. Amer. Math. Soc., 132 (1968), 395–409 | MR | Zbl

[12] A. E. Spencer, “Maximal nonnormal chains in finite groups”, Pacific J. of Math., 27:1 (1968), 167–173 | DOI | MR | Zbl

[13] M. Asaad, “Finite groups some whose $n$-maximal subgroups are normal”, Acta Math. Hung., 54:1–2 (1989), 9–27 | DOI | MR | Zbl

[14] P. Flavell, “Overgroups of second maximal subgroups”, Arch. Math., 64 (1995), 277–282 | DOI | MR | Zbl

[15] X. Y. Guo, K. P. Shum, “Cover-avoidance properties and the structure of finite groups”, J. Pure Appl. Algebra, 181 (2003), 297–308 | DOI | MR

[16] W. Guo, K. P. Shum, A. N. Skiba, “$X$-Semipermutable subgroups of finite groups”, J. Algebra, 315 (2007), 31–41 | DOI | MR | Zbl

[17] B. Li, A. N. Skiba, “New characterizations of finite supersoluble groups”, Sci. China Ser. A: Math., 50:1 (2008), 827–841 | MR

[18] W. Guo, A. N. Skiba, “Finite groups with given $s$-embedded and $n$-embedded subgroups”, J. Algebra, 321 (2009), 2843–2860 | DOI | MR | Zbl

[19] Sh. Li, “Finite non-nilpotent groups all of whose second maximal subgroups are $TI$-groups”, Math. Proc. of the Royal Irish Academy, 100 A:1 (2000), 65–71 | MR

[20] W. Guo, E. V. Legchekova, A. N. Skiba, “Finite groups in which every 3-maximal subgroup commutes with all maximal subgroups”, Math. Notes, 86:3–4 (2009), 325–332 | DOI | MR | Zbl

[21] W. Guo, Yu. V. Lutsenko, A. N. Skiba, “On nonnilpotent groups with every two 3-maximal subgroups permutable”, Siberian Math. J., 50:6 (2009), 988–997 | DOI | MR | Zbl

[22] Yu. V. Lutsenko, A. N. Skiba, “Structure of finite groups with $S$-quasinormal third maximal subgroups”, Ukrainian Math. J., 61:12 (2009), 1915–1922 | DOI | MR | Zbl

[23] Yu. V. Lutsenko, A. N. Skiba, “Finite groups with subnormal second or third maximal subgroups”, Math. Notes, 91:5–6 (2012), 680–688 | DOI | MR | Zbl

[24] D. P. Andreeva, A. N. Skiba, “Finite groups with givem maximal chains of length $\leqslant 3$”, Problems of Physics, Mathematics and Technics, 3 (2011), 39–49 | Zbl

[25] W. Guo, D. P. Andreeva, A. N. Skiba, “Finite groups of Spencer hight $\leqslant 3$”, Algebra Colloquium (to appear)

[26] A. Ballester-Bolinches, L. M. Ezquerro, A. N. Skiba, “On second maximal subgroups of Sylow subgroups of finite groups”, J. Pure Appl. Algebra, 215:4 (2011), 705–714 | DOI | MR | Zbl

[27] V. N. Kniahina, V. S. Monakhov, “On the permutability of $n$-maximal subgroups with Schmidt subgroups”, Trudy Inst. Mat. i Mekh. UrO RAN, 18, no. 3, 2012, 125–130

[28] V. A. Kovaleva, A. N. Skiba, “Finite solvable groups with all $n$-maximal subgroups $\mathfrak{U}$-subnormal”, Sib. Math. J., 54:1 (2013), 65–73 | DOI | MR

[29] V. A. Kovaleva, A. N. Skiba, “Finite soluble groups with all $n$-maximal subgroups $\mathfrak{F}$-subnormal”, J. Group Theory, 17 (2014), 273–290 | DOI | MR

[30] V. S. Monakhov, V. N. Kniahina, “Finite groups with $\mathbb{P}$-subnormal subgroups”, Ricerche di Matematica, 62:2 (2013), 307–322 | DOI | MR

[31] O. H. Kegel, “Zur Struktur mehrfach faktorisierbarer endlicher Gruppen”, Math. Z., 87 (1965), 409–434 | DOI | MR

[32] A. Ballester-Bolinches, L. M. Ezquerro, Classes of Finite Groups, Springer-Verlag, 2006 | MR | Zbl

[33] K. Doerk, “Minimal nicht uberauflosbare, endliche Gruppen”, Math. Z., 91 (1966), 198–205 | DOI | MR | Zbl