Geodesic
On some generalizations of permutability and $S$-permutability
Problemy fiziki, matematiki i tehniki, no. 4 (2013), pp. 47-54.

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Let $H$ and $X$ be subgroups of a finite group $G$. Then we say that $H$ is: $X$-quasipermutable (respectively, $X_S$-quasipermutable) in $G$ provided $G$ has a subgroup $B$ such that $G=N_G(H)B$ and $H$ $X$-permutes with $B$ and with all subgroups (respectively, with all Sylow subgroups) $V$ of $B$ such that $(|H|,|V|)=1$; $X$-propermutable (respectively, $X_S$-propermutable) in $G$ provided $G$ has a subgroup $B$ such that $G=N_G(H)B$ and $H$ $X$-permutes with $B$ and with all subgroups (respectively, with all Sylow subgroups) of $B$. In this paper we analyze the influence of $X$-quasipermutable, $X_S$-quasipermutable, $X$-propermutable and $X_S$-propermutable subgroups on the structure of $G$.
Keywords: finite group, $X$-quasipermutable subgroup, $X_S$-quasipermutable subgroup, $X$-propermutable subgroup, $X_S$-propermutable subgroup, Sylow subgroup, Hall subgroup, $p$-supersoluble group, maximal subgroup, saturated formation, $PST$-group
Mots-clés : $p$-soluble group, $PT$-group. 
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Xiaolan Yi; A. N. Skiba. On some generalizations of permutability and $S$-permutability. Problemy fiziki, matematiki i tehniki, no. 4 (2013), pp. 47-54. http://geodesic.mathdoc.fr/item/PFMT_2013_4_a7/

[1] W. Guo, K. P. Shum, A. N. Skiba, “$X$-semipermutable subgroups of finite groups”, J. Algebra, 215 (2007), 31–41 | DOI | MR

[2] O. Ore, “Contributions in the theory of groups of finite order”, Duke Math. J., 5 (1939), 431–460 | DOI | MR

[3] S. E. Stonehewer, “Permutable subgroups in Infinite Groups”, Math. Z., 125 (1972), 1–16 | DOI | MR | Zbl

[4] O. H. Kegel, “Sylow-Gruppen and Subnormalteiler endlicher Gruppen”, Math. Z., 78 (1962), 205–221 | DOI | MR | Zbl

[5] X. Yi, A. N. Skiba, “Some new characterizations of $PST$-groups”, J. Algebra, 2014 | DOI | MR

[6] X. Yi, A. N. Skiba, “On $S$-propermutable subgroups of finite groups”, Bull. Malays. Math. Sci. Soc. (to appear)

[7] S. Li, Z. Shen, J. Liu, X. Liu, “The influence of $SS$-quasinormality of some subgroups on the structure of finite group”, J. Algebra, 319 (2008), 4275–4287 | DOI | MR | Zbl

[8] Ch. Chen, “Generalization of theorem Srinivasan”, J. Sothwest China Normal Univ., I (1987), 1–4

[9] D. Gorenstein, Finite Groups, Harper Row Publishers, New York–Evanston–London, 1968 | MR | Zbl

[10] H. Su, “Semi-normal subgroups of finite groups”, Math. Mag., 8 (1988), 7–9

[11] S. Li, Z. Shen, J. Liu, X. Liu, “$SS$-quasinormal subgroups of finite group”, Comm. Algebra, 36 (2008), 4436–4447 | DOI | MR | Zbl

[12] V. S. Monakhov, “Finite groups with semi-normal Hall subgroup”, Matem. Zametki, 80:4 (2006), 573–581 | DOI | MR | Zbl

[13] W. Guo, “Finite groups with semi-normal Sylow subgroups”, Acta Math. Sinica. English Series, 24:10 (2008), 1751–1758 | DOI | MR

[14] B. Li, A. N. Skiba, “New characterizations of finite super-soluble groups”, Science in China. Series A, Mathematics, 51 (2008), 827–841 | DOI | MR | Zbl

[15] R. A. Brice, J. Cossey, “The Wielandt subgroup of a finite soluble groups”, J. London Math. Soc., 40 (1989), 244–256 | DOI | MR

[16] J. C. Beidleman, B. Brewster, D. J. S. Robinson, “Criteria for permutability to be transitive in finite groups”, J. Algebra, 222 (1999), 400–412 | DOI | MR | Zbl

[17] A. Ballester-Bolinches, R. Esteban-Romero, “Sylow permutable subnormal subgroups”, J. Algebra, 251 (2002), 727–738 | DOI | MR | Zbl

[18] A. Ballester-Bolinches, J. C. Beidleman, H. Heineken, “Groups in which Sylow subgroups and subnormal subgroups permute”, Illinois J. Math., 47 (2003), 63–69 | MR | Zbl

[19] A. Ballester-Bolinches, J. C. Beidleman, H. Heineken, “A local approach to certain classes of finite groups”, Comm. Algebra, 31 (2003), 5931–5942 | DOI | MR | Zbl

[20] M. Asaad, “Finite groups in which normality or quasinormality is transitive”, Arch. Math., 83:4 (2004), 289–296 | DOI | MR | Zbl

[21] A. Ballester-Bolinches, J. Cossey, “Totally permutable products of finite groups satisfying $SC$ or $PST$”, Monatsh. Math., 145 (2005), 89–93 | DOI | MR

[22] V. O. Lukyanenko, A. N. Skiba, “Finite groups in which $\tau$-quasinormality is a transitive relation”, Rend. Semin. Univ. Padova, 124 (2010), 231–246 | DOI | MR | Zbl

[23] J. C. Beidleman, M. F. Ragland, “Subnormal, permutable, and embedded subgroups in finite groups”, Central Eur. J. Math., 9:4 (2011), 915–921 | DOI | MR | Zbl

[24] A. Ballester-Bolinches et al., “Finite solvable groups in which seminormality is a transitive relation”, Beitr. Algebra Geom. | DOI

[25] A. Ballester-Bolinches, J. C. Beidleman, A. D. Feldman, “Some new characterizations of solvable PST-groups”, Ricerche mat. | DOI

[26] Y. Li, L. Wang, Y. Wang, “Finite groups in which $(S)$-semipermutability permutability is a transitive relation”, Internat. J. Algebra, 2:3 (2008), 143–152 | MR | Zbl

[27] K. Al-Sharo et al., “Some Characterizations of Finite Groups in which Semipermutability is a Transitive Relation”, Forum Math., 22 (2010), 855–862 | DOI | MR | Zbl

[28] W. Guo, A. N. Skiba, “Criterions of Existence of Hall Subgroups in Non-soluble Finite Groups”, Acta Math. Sinica, 26:2 (2010), 295–304 | DOI | MR | Zbl

[29] W. Guo, A. N. Skiba, “New criterions of existence and conjugacy of Hall subgroups of finite groups”, Proc. Amer. Math. Soc., 139 (2011), 2327–2336 | DOI | MR | Zbl

[30] X. B. Wei, X. Y. Guo, “On $SS$-quasinormal subgroups and the structure of finite groups”, Science China Mathematics, 54:3 (2011), 449–456 | DOI | MR | Zbl

[31] B. Huppert, Endliche Gruppen, v. I, Springer-Verlag, Berlin–Heidelberg–New York, 1967 | MR | Zbl

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

[33] A. Ballester-Bolinches, R. Esteban-Romero, M. Asaad, Products of Finite Groups, Walter de Gruyter, Berlin–New York, 2010 | MR | Zbl

[34] G. Zacher, “I gruppi risolubili finiti in cui i sottogruppi di composizione coincidono con i sottogrupi quasi-normali”, Atti Accad. Naz. Lincei Rend. cl. Sci. Fis. Mat. Natur., 8:37 (1964), 150–154 | MR

[35] R. K. Agrawal, “Finite groups whose subnormal subgroups permute with all Sylow subgroups”, Proc. Amer. Math. Soc., 47 (1975), 77–83 | DOI | MR | Zbl

[36] D. J. S. Robinson, “The structure of finite groups in which permutability is a transitive relation”, J. Austral. Math. Soc., 70 (2001), 143–159 | DOI | MR | Zbl

[37] X. Yi, A. N. Skiba, “Propermutable characterizations of soluble $PT$-groups and $PST$-groups”, Science China. Mathematics (to appear)

[38] L. A. Shemetkov, Formations of Finite Groups, Nauka, M., 1978 | MR | Zbl

[39] D. J. S. Robinson, A Course in the Theory of Groups, Springer-Verlag, New-York–Heidelberg–Berlin, 1982 | MR | Zbl

[40] V. V. Podgornaya, “Seminormal subgroups and supersolubility of finite groupds”, Vesti NAN Belarus. Ser. Phys.-Math. Sciences, 4 (2000), 22–26 | MR

[41] A. N. Skiba, “On the $\mathfrak{F}$-hypercentre and the intersection of all $\mathfrak{F}$-maximal subgroups of a finite group”, J. Pure Appl. Algebra, 216:4 (2012), 789–799 | DOI | MR | Zbl

[42] Y. Li, Y. Wang, “The influence of $\pi$-quasinormality of some subgroups of a finite group”, Arch. Math., 81 (2003), 245–252 | DOI | MR | Zbl