Geodesic
Finite Groups with Some Maximal Subgroups of Sylow Subgroups $\mathscr M$-Supplemented
Matematičeskie zametki, Tome 86 (2009) no. 5, pp. 692-704 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A subgroup $H$ of a group $G$ is said to be $\mathscr M$‐supplemented in $G$ if there exists a subgroup $B$ of $G$ such that $G=HB$ and $TB for every maximal subgroup $T$ of $H$. In this paper, we obtain the following statement: Let $\mathscr F$ be a saturated formation containing all supersolvable groups and $H$ be a normal subgroup of $G$ such that $G/H\in\mathscr F$. Suppose that every maximal subgroup of a noncyclic Sylow subgroup of $F^{*}(H)$, having no supersolvable supplement in $G$, is $\mathscr M$-supplemented in $G$. Then $G\in\mathscr F$.
Keywords: Sylow subgroup, $\mathscr M$-supplemented subgroup, finite group, Hall subgroup, Fitting subgroup, $p$-nilpotent group.
Mots-clés : formation, supersolvable group 
@article{MZM_2009_86_5_a6,
     author = {Long Miao},
     title = {Finite {Groups} with {Some} {Maximal} {Subgroups} of {Sylow} {Subgroups} $\mathscr M${-Supplemented}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {692--704},
     year = {2009},
     volume = {86},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2009_86_5_a6/}
}
TY  - JOUR
AU  - Long Miao
TI  - Finite Groups with Some Maximal Subgroups of Sylow Subgroups $\mathscr M$-Supplemented
JO  - Matematičeskie zametki
PY  - 2009
SP  - 692
EP  - 704
VL  - 86
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/MZM_2009_86_5_a6/
LA  - ru
ID  - MZM_2009_86_5_a6
ER  - 
%0 Journal Article
%A Long Miao
%T Finite Groups with Some Maximal Subgroups of Sylow Subgroups $\mathscr M$-Supplemented
%J Matematičeskie zametki
%D 2009
%P 692-704
%V 86
%N 5
%U http://geodesic.mathdoc.fr/item/MZM_2009_86_5_a6/
%G ru
%F MZM_2009_86_5_a6
Long Miao. Finite Groups with Some Maximal Subgroups of Sylow Subgroups $\mathscr M$-Supplemented. Matematičeskie zametki, Tome 86 (2009) no. 5, pp. 692-704. http://geodesic.mathdoc.fr/item/MZM_2009_86_5_a6/

[1] S. Srinivasan, “Two sufficient conditions for supersolvability of finite groups”, Israel J. Math., 35:3 (1980), 210–214 | DOI | MR | Zbl

[2] M. Asaad, M. Ramadan, A. Shaalan, “Influence of $\pi$-quasinormality on maximal subgroups of Sylow subgroups of Fitting subgroup of a finite group”, Arch. Math. (Basel), 56:6 (1991), 521–527 | DOI | MR | Zbl

[3] M. Asaad, “On maximal subgroups of Sylow subgroups of finite groups”, Comm. Algebra, 26:11 (1998), 3647–3652 | DOI | MR | Zbl

[4] Y. Wang, “Finite groups with some subgroups of Sylow subgroups $c$-supplemented”, J. Algebra, 224:2 (2000), 467–478 | DOI | MR | Zbl

[5] L. Miao, W. Guo, “Finite groups with some primary subgroups $\mathscr F$-s-supplemented”, Comm. Algebra, 33:8 (2005), 2789–2800 | DOI | MR | Zbl

[6] W. Guo, The Theory of Classes of Groups, Math. Appl., 505, Kluwer Acad. Publ., Dordrecht, 2000 | MR | Zbl

[7] D. J. S. Robinson, A Course in the Theory of Groups, Grad. Texts in Math., 80, Springer-Verlag, New York, NY, 1993 | MR | Zbl

[8] M. Xu, An Introduction to Finite Groups, Sci. Press, Beijing, 1999

[9] A. N. Skiba, “A note on $c$-normal subgroups of finite groups”, Algebra Discrete Math., 2005, no. 3, 85–95 | MR | Zbl

[10] B. Huppert, Endliche Gruppen. I, Grundlehren Math. Wiss., 134, Springer-Verlag, Berlin, 1967 | MR | Zbl

[11] F. Gross, “Conjugacy of odd order Hall subgroups”, Bull. London Math. Soc., 19:4 (1987), 311–319 | DOI | MR | Zbl

[12] B. Huppert, N. Blackburn, Finite Groups. III, Grundlehren Math. Wiss., 243, Springer-Verlag, Berlin, 1982 | MR | Zbl

[13] Y. Wang, H. Wei, Y. Li, “A generalisation of Kramer's theorem and its applications”, Bull. Austral. Math. Soc., 65:3 (2002), 467–475 | DOI | MR | Zbl