Geodesic
Finite groups with given local sections
Problemy fiziki, matematiki i tehniki, no. 3 (2019), pp. 107-110.

Voir la notice de l'article provenant de la source Math-Net.Ru

A group is called primary if it is a finite $p$-group for some prime $p$. If $\sigma=\{\sigma_i\mid i\in I\}$ is some partition of $\mathbb{P}$, that is, $P=\bigcup_{i\in I}\sigma_i$ and $\sigma_i\cap\sigma_j=\varnothing$ for all $i\ne j$, then we say that a finite group $G$ is: $\sigma$-primary if it is a $\sigma_i$-group for some $i$; $\sigma$-nilpotent if $G=G_1\times\dots\times G_n$ for some $\sigma$-primary groups $G_1,\dots,G_n$. If $N=N_G(A)$ for some primary non-identity subgroup $A$ of $G$, then we say that $N/A_G$ is a local section of $G$. In this paper, we study a finite group $G$ under hypothesis that all proper local sections of $G$ belong to a saturated hereditary formation $\mathfrak{F}$, and we determine the normal structure of $G$ in the case when all local sections of $G$ are $\sigma$-nilpotent.
Keywords: finite group, hereditary saturated formation, $\mathfrak{F}$-hypercentre, local section, $\sigma$-nilpotent group. 
@article{PFMT_2019_3_a18,
     author = {B. Hu and J. Huang and A. N. Skiba},
     title = {Finite groups with given local sections},
     journal = {Problemy fiziki, matematiki i tehniki},
     pages = {107--110},
     publisher = {mathdoc},
     number = {3},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PFMT_2019_3_a18/}
}
TY  - JOUR
AU  - B. Hu
AU  - J. Huang
AU  - A. N. Skiba
TI  - Finite groups with given local sections
JO  - Problemy fiziki, matematiki i tehniki
PY  - 2019
SP  - 107
EP  - 110
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PFMT_2019_3_a18/
LA  - en
ID  - PFMT_2019_3_a18
ER  - 
%0 Journal Article
%A B. Hu
%A J. Huang
%A A. N. Skiba
%T Finite groups with given local sections
%J Problemy fiziki, matematiki i tehniki
%D 2019
%P 107-110
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PFMT_2019_3_a18/
%G en
%F PFMT_2019_3_a18
B. Hu; J. Huang; A. N. Skiba. Finite groups with given local sections. Problemy fiziki, matematiki i tehniki, no. 3 (2019), pp. 107-110. http://geodesic.mathdoc.fr/item/PFMT_2019_3_a18/

[1] A.N. Skiba, “ On $\sigma$-subnormal and $\sigma$-permutable subgroups of finite groups”, J. Algebra, 436 (2015), 1–16 | DOI | MR | Zbl

[2] L.A. Shemetkov, Formations of finite groups, Nauka, M., 1978, 272 pp. | MR | Zbl

[3] A. Ballester-Bolinches, K. Doerk, M.D. Perez-Ramos, “On the lattice of $\mathfrak{F}$-subnormal subgroups”, J. Algebra, 148 (1992), 42–52 | DOI | MR | Zbl

[4] A.F. Vasil'ev, A.F. Kamornikov, V.N. Semenchuk, “On lattices of subgroups of finite groups”, Infinite groups and related algebraic structures, Institut Matematiki AN Ukrainy, Kiev, 1993, 27–54 | MR | Zbl

[5] A. Ballester-Bolinches, L.M. Ezquerro, Classes of Finite Groups, Springer-Verlag, Dordrecht, 2006, 385 pp. | MR | Zbl

[6] V.A. Kovaleva, “A criterion for a finite group to be $\sigma$-soluble”, Commun. Algebra, 46 (2019), 5410–5415 | DOI | MR

[7] A.N. Skiba, “Some characterizations of finite $\sigma$-soluble $P\sigma T$-groups”, J. Algebra, 495 (2018), 114–129 | DOI | MR | Zbl

[8] J.C. Beidleman, A.N. Skiba, “On $\tau_\sigma$-quasinormal subgroups of finite groups”, J. Group Theory, 20:5 (2017), 955–964 | DOI | MR

[9] A.N. Skiba, “A generalization of a Hall theorem”, J. Algebra Appl., 15:4 (2015), 21–36 | MR

[10] K. Doerk, T. Hawkes, Finite Soluble Groups, Walter de Gruyter, Berlin–New York, 1992, 891 pp. | MR

[11] L.A. Shemetkov, A.N. Skiba, Formations of Algebraic Systems, Nauka, M., 1989, 254 pp. | MR | Zbl

[12] A.N. Skiba, B. Hu, J. Huang, On generalized local subgroups of finite groups, Preprint, 2019, 18 pp. | MR

[13] R. Baer, “The influence on a finite groups of certain types of its proper subgroups”, Illinois J. Math., 1 (1957), 115–187 | DOI | MR | Zbl

[14] V. Fedri, U. Tiberio, “Sui gruppi finiti i sui sottogruppi locali propri sono superssolubili”, Boll. Unione Mat. Ital., 17:1 (1980), 73–78 | MR | Zbl

[15] J.C. Beidleman, “The influence ons a finite groups of certain types of its proper subgroups”, Riv. Mat. Univ. Parma, 9:2 (1968), 247–260 | MR | Zbl

[16] Z. Chi, A.N. Skiba, “On semi-$\sigma$-nilpotent finite groups”, Journal of Algebra and Its Applications, 2019 | DOI | MR

[17] V.I. Gorbachev, “Local Fsubgroups of finite groups”, Problems in Algebra, 1986, no. 2, 62–72 | MR | Zbl

[18] Chih-Han Sah, “On a generalization of finite nilpotent groups”, Math. Z., 68:1 (1957), 189–204 | DOI | MR | Zbl

[19] M. Weinstein, Between Nilpotent and Solvable, Polygonal Publishing House, Passaic, 1982, 232 pp. | MR | Zbl

[20] V.S. Monakhov, I.L. Sokhor, “On cofactors of subnormal subgroups”, Journal of Algebra and Its Applications, 15:9 (2016) | DOI | MR

[21] V.S. Monakhov, I.L. Sokhor, “On Groups with Formational Subnormal Sylow Subgroups”, J. Group Theory, 21:2 (2018), 273–287 | DOI | MR | Zbl

[22] N.M. Adarchenko, I.G. Blistets, V.N. Rizhik, “On finite semi-$p$-decomposable groups”, Problems of Physics, Mathematics and Technics, 34:1 (2018), 41–44 | MR | Zbl

[23] N.S. Kosenok, V.M. Selkin, V.N. Rizhik, V.N. Mitsik, “On finite semi-$\pi$-special groups Mitsik”, Problems of Physics, Mathematics and Technics, 39:2 (2019), 88–91 | MR