Author
Zhencai Shen
Other affiliations: Peking University, Soochow University (Suzhou)
Bio: Zhencai Shen is an academic researcher from China Agricultural University. The author has contributed to research in topics: Locally finite group & Finite group. The author has an hindex of 4, co-authored 11 publications receiving 51 citations. Previous affiliations of Zhencai Shen include Peking University & Soochow University (Suzhou).
Papers
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TL;DR: In this paper, Baer and Wielandt considered the intersection of the normalizers of all subgroups of a finite group and showed that the subgroup S ∞ (G ) is solvable if and only if the residual residual G N is nilpotent.
22 citations
TL;DR: A subgroup H of a finite group G is said to be quasinormally embedded in G if for every Sylow subgroup P of H, there is a quasinormal (resp. S-quasinormal) subgroup K in G such that every member in some ℳ d (P) is also a Sylow Subgroup of K as discussed by the authors.
Abstract: Abstract A subgroup H of the finite group G is said to be quasinormally (resp. S-quasinormally) embedded in G if for every Sylow subgroup P of H, there is a quasinormal (resp. S-quasinormal) subgroup K in G such that P is also a Sylow subgroup of K. Groups with certain quasinormally (resp. S-quasinormally) embedded subgroups of prime-power order are studied. For example, if a group G has a normal subgroup H such that G/H ∈ ℱ and such that for each Sylow subgroup P of H, every member in some ℳ d (P) is quasinormally embedded in G, then G ∈ ℱ: here ℳ d (P) is a set of maximal subgroups of P with intersection the Frattini subgroup.
11 citations
TL;DR: In this paper, it was shown that if G.G.G is a non-cyclic subgroup, then the subgroup A(G) is the intersection of the normalizers of all noncyclic groups of G.
Abstract: G, we dene the subgroup A(G) to be intersection of the normalizers of all non-cyclic subgroups of G. Set A0 = 1. Dene Ai+1(G)=Ai(G) = A(G=Ai(G)) for i 1. By A1 (G) denote the terminal term of the ascending series. It is proved that if G.
9 citations
TL;DR: In this article, the intersection of the normalizers of all non-cyclic subgroups of a finite group G is studied and the results of Passman, Bozikov, and Janko are extended to non-nilpotent finite groups.
Abstract: Baer and Wielandt in 1934 and 1958, respectively, considered that the intersection of the normalizers of all subgroups of G and the intersection of the normalizers of all subnormal subgroups of G. In this article, for a finite group G, we define the subgroup S(G) to be intersection of the normalizers of all non-cyclic subgroups of G. Groups whose noncyclic subgroups are normal are studied in this article, as well as groups in which all noncyclic subgroups are normalized by all minimal subgroups. In particular, we extend the results of Passman, Bozikov, and Janko to non-nilpotent finite groups.
7 citations
TL;DR: In this paper, the structure of a finite group G under some assumptions on the S-quasinormally embedded or SSquasinormal subgroups in ℳ¯¯¯¯ d� (P), for each prime p, and Sylow p-subgroups P of G is studied.
Abstract: Let d be the smallest generator number of a finite p-group P, and let ℳ
d
(P) = {P
1,..., P
d
} be a set of maximal subgroups of P such that ∩
=1
P
i
=Φ(P). In this paper, the structure of a finite group G under some assumptions on the S-quasinormally embedded or SS-quasinormal subgroups in ℳ
d
(P), for each prime p, and Sylow p-subgroups P of G is studied.
6 citations
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01 Jan 2010
Abstract: 1. (i) Suppose K is a conjugacy class of Sn contained in An; then K is called split if K is a union of two conjugacy classes of An. Show that the number of split conjugacy classes contained in An is equal to the number of characters χ ∈ Irr(Sn) such that χAn is not irreducible. (Hint. Consider the vector space of class functions on An which are invariant under conjugation by the transposition (12).)
613 citations
TL;DR: In this article, a framework for generalisations of Baer's norm has been given for a class of finite nilpotent groups, where the C -norm κ C (G ) of a finite group G is defined as the intersection of the normalisers of the subgroups of G not in C.
16 citations
TL;DR: In this article, the authors studied the relationship between G = N F ∞ (G ) and G ∈ F ǫ (G), and the formations F such that G ∆ F would imply that G = nF ∞(G ), and the formation G such that g ∈ G would imply N F∞ ( G ), which is the case for all groups whose F -residuals are nilpotent.
12 citations
TL;DR: In this article, the relationship between the π F -norm and the δ F -hypercentre of a finite group G was investigated. But the relationship was not studied in the context of finite π-groups.
8 citations
8 citations