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Fin.te groups whose n.maximal subgroups are σ-subn.rmal

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Scien.e Chin. Mathematics 2019年第7期62卷 1355-1372页

作者： Wen.in.Guo alexander n. skiba School of Mathematical Scien.es University of Science and Technology of China Hefei 230026 China Departmen. of Mathematics an. Techn.logies of Programmin. Francisk Skorina Gomel State University Gomel 246019 Belarus

Let σ={σi | i ∈ I} be some partition.of the set of all primes P. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member ≠ 1 of H is a Hall σi-subgroup of G, for some i ∈ I, an. H con.ain. exactly on. Hall σi-subgroup of G for every σi ∈σ(G). A subgroup H of G is said to be:σ-permutable or σ-quasin.rmal in.G if G possesses a complete Hall σ-set H such that HAx= AxH for all A ∈ H an. x ∈ G:σ-subn.rmal in.G if there is a subgroup chain.A = A0≤A1≤···≤ At = G such that either Ai-1■Ai or Ai/(Ai-1)Ai is a fin.te σi-group for some σi ∈σ for all i = 1,..., *** Mn.n.1 n. then.Mn.is said to be an.n.maximal subgroup of G. If each n.maximal subgroup of G is σ-subn.rmal(σ-quasin.rmal,respectively) in.G but, in.the case n.> 1, some(n.1)-maximal subgroup is n.t σ-subn.rmal(n.t σ-quasin.rmal,respectively) in.G, we write mσ(G)= n.mσq(G)= n. respectively).In.this paper, we show that the parameters mσ(G) an. mσq(G) make possible to boun. the σ-n.lpoten. len.th lσ(G)(see below the defin.tion. of the terms employed), the ran. r(G) an. the n.mber |π(G)| of all distin.t primes dividin. the order |G| of a fin.te soluble group G. We also give the con.ition. un.er which a fin.te group is σ-soluble or σ-n.lpoten., an. describe the structure of a fin.te soluble group G in.the case when.mσ(G)=|π(G)|. Some kn.wn.results are gen.ralized.

关键词： fin.te group n.maximal subgroup σ-subn.rmal subgroup σ-quasin.rmal subgroup σ-soluble group σ-n.lpoten. group

On.σ-semipermutable Subgroups of Fin.te Groups

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Acta Mathematica Sin.ca,En.lish Series 2018年第9期34卷 1379-1390页

作者： Wen.Bin.GUO alexander n. skiba Departmen. of Mathematics University of Science and Technology of China Hefei 230026 P. R. China Departmen. of Mathematics an. Techn.logies of Programmin. Francisk Skorina Gomel State University Gomel 246019 Belarus

Let a = {σi｜ i ∈ I} be some partition.of the set of all primes P, G a fin.te group an. σ（G） = {σi｜σi ∩ π （G） ≠ Ф}. A set H of subgroups of G is said to be a complete Hall or-set of G if every member ≠ 1 of H is a Hall σi-subgroup of G for some σi ∈ σ an. H con.ain. exactly on. Hall σi-subgroup of G for every σi ∈ σ（G）. A subgroup H of G is said to be： σ-semipermutablc in.G with respect to H if HHi x = Hi x H for all x ∈ G an. all x ∈ G an. all Hi ∈H such that （｜H｜, ｜Hi｜） = 1; σ-semipermutable in.G if H is σ-semipermutable in.G with respect to some complete Hall σ-set of G. We study the structure of G bein. based on.the assumption.that some subgroups of G are σ-semipermutable in.G.

关键词： Fin.te group Hall subgroup p-soluble group p-supersoluble group σ-semipermutable subgroup

Fin.te Groups of Spen.er Height ≤ 3

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Algebra Colloquium 2015年第3期22卷 437-444页

作者： Wen.in.Guo Din. P. An.reeva alexander n. skiba School of Mathematical Scien.esUn.versity of Scien.e an. Techn.logy of Chin. Hefei 230026 China Departmen. of Mathematics Francisk Skorina Gomel State University Gomel 246019 Belarus

In.this paper, we describe the fin.te groups G in.which every maximal chain.M3 〈 M2 〈 M1 〈 M0 = G con.ain. a proper subn.rmal subgroup of G.

关键词： fin.te group subn.rmal subgroup maximal chain Spen.er height Schmidt group

A gen.ralized CAP-subgroup of a fin.te group

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Scien.e Chin. Mathematics 2015年第10期58卷 2133-2144页

作者： GUO Wen.in.skiba alexander n. YAn. n.n.in. School of Mathematical Scien.es University of Science and Technology of China Departmen. of Mathematics Francisk Skorina Gomel State University School of Scien.e Jiangnan University

Let A be a subgroup of a fin.te group G. We say that A is a gen.ralized CAP-subgroup of G if for each chief factor H/K of G either A avoids H/K or the followin. holds:(1) If H/K is n.n.abelian. then.H :(A ∩H)K | is a p′-n.mber for every p ∈π((A ∩H)K/K);(2) If H/K is a p-group, then.|G : n.(K(A ∩H))| is a p-n.mber. In.this paper, we use the gen.ralized CAP-subgroup to characterize the structure of fin.te *** n.w characterization. of the hypercyclically embedded subgroups of a fin.te group are obtain.d an. a series of kn.wn.results are gen.ralized.

关键词： fin.te group gen.ralized CAP-subgroup U-embedded subgroup solubly saturated formation p-n.lpoten. group supersoluble subgroup

On.solubility an. supersolubility of some classes of fin.te groups

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Scien.e Chin. Mathematics 2009年第2期52卷 272-286页

作者： SHUM Kar Pin. skiba alexander n. Departmen. of Mathematics University of Hong Kong Departmen. of Mathematics Francisk Skorina Gomel State University

AbstractLet G be a fin.te group an. H a subgroup of G. Then.H is said to be S-permutable in.G if HP = PH for all Sylow subgroups P of G. Let H sG be the subgroup of H gen.rated by all those subgroups of H which are S-permutable in.G. Then.we say that H is S-embedded in.G if G has a n.rmal subgroup T an. an.S-permutable subgroup C such that T ∩ H ? H sG an. HT = C.Our main.result is the followin.Theorem AA group G is supersoluble if an. on.y if for every n.n.cyclic Sylow subgroup P of the gen.ralized Fittin. subgroup F*(G) of G, at least on. of the followin. holds: Every maximal subgroup of P is S-embedded in.G.Every cyclic subgroup H of P with prime order or order 4 (if P is a n.n.abelian.2-group an. H ? Z ∞ (G)) is S-embedded in.G.

关键词： S-permutable subgroups S-embedded subgroups saturated formation. gen.ralized Fittin. subgroups supersoluble groups 20D10 20D20 20D25

n.w characterization. of fin.te supersoluble groups

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Scien.e Chin. Mathematics 2008年第5期51卷 827-841页

作者： alexander n.skiba Departmen. of Mathematics Gomel State Un.versity of F.Skorin.

Let A be a subgroup of a group G an. X a n.n.mpty subset of G. A is called an.X-semipermutable subgroup of G if A has a supplemen. T in.G such that for every subgroup T 1 of T there exists an.elemen. x ∈ X such that ... 详细信息

关键词： fin.te groups supplemen. 2-maximal subgroups X-semipermutable supersoluble groups 20D10

Fin.te groups with systems of Σ-embedded subgroups

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Scien.e Chin. Mathematics 2011年第9期54卷 1909-1926页

作者： GUO Wen.in.skiba alexander n Departmen. of Mathematics University of Science and Technology of China Hefei 230026 China Departmen. of Mathematics Francisk Skorina Gomel State University Gomel 246019 Belarus

In.recen. years,a series of papers about cover-avoidin. property of subgroups appeared an. all the studies were con.ected with chief factors of a fin.te ***,about the cover-avoidin. property of subgroups for n.n.chief factor,there is n. study up to *** purpose of this paper is to build the *** A be a subgroup of a fin.te group G an. Σ:G0≤G1≤…≤Gn.some subgroup series of *** that for each pair(K,H) such that K is a maximal subgroup of H an. G i 1 K n.*** this paper,we study the fin.te groups with given.systems of Σ-embedded *** basic properties of Σ-embedded subgroups are established an. some n.w characterization. of some classes of fin.te groups are given.an. some kn.wn.results are gen.ralized.

关键词： fin.te group ∑-embedded subgroups ∑p-embedded subgroup soluble group supersoluble group p-n.lpoten. group saturated formation

Criteria of Existen.e of Hall Subgroups in.n.n.soluble Fin.te Groups

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Acta Mathematica Sin.ca,En.lish Series 2010年第2期26卷 295-304页

作者： Wen.Bin.GUO A. n. skiba Departmen. of Mathematics Xuzhou Normal University Xuzhou 221116 P. R. China Departmen. of Mathematics Francisk Skorina Gomel State University Gomel 246019 Belarus

In.this paper, the famous Hall theorem an. the famous Schur-Zassen.aus theorem are gen.ralized.

关键词： fin.te groups n.n.soluble groups Hall subgroups

On.Existen.e of Hall Subgroups in.Fin.te Groups

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Algebra Colloquium 2017年第1期24卷 75-82页

作者： Yufen. Liu Wen.in.Guo n. skiba School of Mathematics an. In.ormation.l Scien.eShan.on. In.titute of Busin.ss an. Techn.logy Yantai Shandong 264005 China School of Mathematical Scien.es University of Science and Technology of ChinaHefei 230026 China Departmen. of Mathematics Gomel State University of F. Skorina Gomel 246028 Belarus

In.this paper, we give some n.w con.ition. of the existen.e of Hall subgroups in.n.n.soluble fin.te groups, an. so the famous Hall theorem an. Schur-Zassen.aus theorem are gen.ralized.

关键词： fin.te groups n.n.soluble groups Hall subgroups