Geodesic
On one generalization of Baer's theorems about hypercenter and nilpotent residual
Problemy fiziki, matematiki i tehniki, no. 3 (2013), pp. 84-88.

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Let $\mathfrak{F}$ be a class of finite groups which are the direct products of their Hall $\pi_i$-subgroups corresponding to a given partition $\sigma=\{\pi_i|i\in I, i\ne j\rightarrow\pi_i\cap\pi_j=\varnothing\}$ of a nonempty subset $\pi$ of the set of all primes. This class is a local formation. In this paper the properties of $\mathfrak{F}$-hypercenter and $\mathfrak{F}$-residual of a finite group are studied. It was shown that for a finite $\pi$-group $G$ the intersection of all normalizers of all maximal $\pi_i$-subgroups for all $i$ is the $\mathfrak{F}$-hypercenter of $G$. As corollaries were obtained well-known properties of hypercenter and nilpotent residual of finite groups.
Keywords: finite group, formation of finite groups, $\mathfrak{F}$-hypercenter, $\mathfrak{F}$-residual.
Mots-clés : local formation 
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V. I. Murashka. On one generalization of Baer's theorems about hypercenter and nilpotent residual. Problemy fiziki, matematiki i tehniki, no. 3 (2013), pp. 84-88. http://geodesic.mathdoc.fr/item/PFMT_2013_3_a13/

[1] R. Baer, “Group Elements of Prime Power Index”, Trans. Amer. Math. Soc., 75:1 (1953), 20–47 | DOI | MR | Zbl

[2] A. F. Vasil'ev, S. 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 (In Russian) | MR

[3] A. F. Vasil'ev, S. F. Kamornikov, “On the Kegel–Shemetkov problem on lattices of generalized subnormal subgroups of finite groups”, Algebra Logika, 41:4 (2002), 411–428 | MR

[4] K. Doerk, T. Hawkes, Finite soluble groups, Walter de Gruyter, Berlin–New York, 1992, 891 pp. | MR