On θ∗-completions of F-abnormal Subgroups

(School of Science,Dalian Ocean University,Dalian116023,China)

On θ∗-completions of F-abnormal Subgroups

GAO Hui,GAO Sheng-zhe,YIN Li

(School of Science,Dalian Ocean University,Dalian116023,China)

LetFbe a saturated formation of fnite groups.Given a proper subgroupHof a groupG,a subgroupHofGis calledF-normal inGifG/HGbelongs toF;otherwiseHis said to beF-abnormal inG.In this paper,we investigate the structure of a fnite group byθ∗-completions ofF-abnormal subgroups.

θ∗-completions;maximal subgroups;2-maximal subgroups;F-abnormal subgroups

§1.Introduction

All groups considered are assumed to be fnite.Given a groupG,letH＜Gdenote thatHis a proper subgroup ofGand writeHG=coreG(H),the core ofHinG,the largest normal subgroup ofGwhich is contained inH.LetFbe a saturated formation of the groups andUdenote the class of all supersolvable groups.TheF-residual ofGis denoted byGF,the intersection of all normal subgroupsNofGsatisfyingG/N∈F.For any subgroupCofG,letk(C)denote the subgroup generated by all proper subgroups ofCwhich are normal inG.

Given a maximal subgroupMof a groupG,aθ-completion ofMinGis any subgroupCofGsuch thatCis not contained inMwhileMG,the core ofMinG,is contained inCandC/MGhas no proper normal subgroup ofG/MG.The concept ofθ-completions for maximal subgroups was introduced in[1]by Zhao Yaoqing,and was used to investigate the structure ofG.On this topic,there exist a series of papers of the research,for example,see[1-2].In order to generalizeθ-completions for maximal subgroups,the concept ofθ∗-completions for general subgroups is introduced in this paper.

LetFbe a saturated formation of the groups which containsU.The groupsGinFcan be characterized by the minimal subgroups ofGFand byF-abnormal maximal subgroups, for example,see[34].The concept ofF-abnormal maximal subgroups was introduced in[5].A maximal subgroupMofGis calledF-normal inGifG/MG∈F;otherwiseMis said to beF-abnormal inG.In[4],the groupGinFwas characterized by Deskins completions ofF-abnormal maximal subgroups.The main results were following.

Theorem A[4,Theorem2.1]LetFbe a saturated formation which containsU.Then the fnite groupGbelongs toFif and only if,for everyMinand|G:M|a composite},Mhas a maximal completionCsuch thatG=CMandC/k(C)has square free order.

Theorem B[4,Theorem2.2]LetFbe a saturated formation which containsUand letGbe a fnite group which isS4-free.ThenGbelongs toFif and only if every memberMofUc(G)has a maximal completionCsuch thatC/k(C)is cyclic with|C/k(C)|≥|G:M|.

In the present paper,we defne the concepts ofF-abnormal subgroups andθ∗-completions for proper subgroups.At the same time,the groupGinFcan be characterized by means of theθ∗-completion ofF-abnormal subgroups.

All unexplained notation and terminology are standard in this paper.

Defnition 1.1Given a proper subgroupHof a groupG,we callCaθ∗-completion ofHinGif

(1)

(2)IfC1/HGis a proper subgroup ofC/HGandC1/HG◁G/HG,thenG/=〈H,C1〉.

Denote byθI(H)the set of allθ∗-completions ofHinG.Cis said to be maximal inθI(H) if there is no elementDinθI(H)such thatC＜D.

Defnition 1.2A subgroupHofGis calledF-normal inGifG/HG∈F;otherwiseHis said to beF-abnormal inG.

Defnition 1.3For convenience,we introduce the following notations.

UisF-abnormal inG}.

Ucand|G:M|a composite}.

Fand|G:H|a composite}.

IfF(G)/=Ø,we defne Φ1(G)=∩{M|M∈F(G)};otherwise Φ1(G)=G.

Fc

§2.Preliminaries

Lemma 2.1Given a proper subgroupHof a groupG,ifCis a maximalθ∗-completion forH,N≤HandthenC/Nis a maximalθ∗-completion forH/N.Conversely,ifC/Nis a maximalθ∗-completion forH/N,thenCis a maximalθ∗-completion forH.

Lemma 2.2[6,Lemma2.1]LetFbe a formation andGa group.IfG/∈F,then there exists a normal subgroupNofGsuch thatG/N∈b(F),theQ-boundary ofF,i.e.,G/N/∈F, but every proper homomorphic image ofG/Nbelongs toF.Furthermore,G/Npossesses a unique minimal normal subgroup.

Lemma 2.3LetK/Nbe the unique minimal normal subgroup ofG/N.H/Nis a proper subgroup ofG/NandthenHG=N.

ProofSupposeN＜HG,then 1/=HG/N◁G/N.BecauseK/Nis the unique minimal normal subgroup ofG/N,we haveK/N⊆HG/N.Moreover,K/N⊆H/N,a contradiction.

Lemma 2.4LetHbe a proper subgroup of a groupGandCbe a maximalθ∗-completion ofH.IfNis the unique minimal normal subgroup ofGandthenCis a maximal subgroup ofCN.

ProofEvidently,C≤CN.IfC=CN,i.e.,N≤C,a contradiction.It follows thatC＜CN.LetBbe a maximal subgroup ofCNsuch thatBecauseCis a maximalθ∗-completion ofH,we have thatBis not aθ∗-completion ofH.Noticing thatG=〈H,B〉, andHG⊆B,so there exists a normal subgroupLofGsuch thatHG≤L＜BandLis aθ∗-completion ofH.The uniqueness ofNimplies that.This yeilds thatCN≤B,a contradiction.

Lemma 2.5[7,Lemma4.9]LetMbe a solvable maximal subgroup ofG.Kis a nonsolvable normal subgroup ofGsuch thatG=MK,thenM∩K＞1.

Lemma 2.6LetGbe a non-solvable group andHa proper subgroup ofG.Assume thatGhas a unique minimal normal subgroupNandNis non-solvable.IfCis a maximalθ∗-completion ofHsuch thatChas Sylow towers,thenCis a maximal subgroup ofG.

ProofIfN≤C,sinceChas Sylow towers,Nis solvable,a contradiction.We thus may assume thatConsider the subgroup.By Lemma 2.4,we haveBy Lemma 2.5,it concludes that

WriteB=C∩N.ThenB◁CandBhas Sylow towers.Letpbe the largest prime divisor dividing|B|andP∈Sylp(B),thenSincePcharthenbutOtherwise,P◁N,so 1/=P≤F(N)＜N.In particular,F(N)charwe conclude thatF(N)◁G,which is contrary to the fact thatNis the unique minimal normal subgroup ofG. ThusC≤NE(P)＜E.SinceCis maximal inE,we have thatC=NE(P).Moreover,this yieldsNN(P)=NE(P)∩N=C∩N=B.We claim thatP∈Sylp(N).Otherwise,there existsP∗∈Sylp(N)such thatP＜P∗,so we havea contradiction.Applying the Frattini argument,we see thatG=NG(P)N.

On the other hand,sinceNG(P)＜G,there exists a maximal subgroupLofGwhich containsNG(P)such thatG=LN.Obviously,C=NE(P)≤NG(P)≤L.We assert thatC=L.Suppose thatC/=L.ThenCis a proper subgroup ofL.SinceCis a maximalθ∗-completion ofH,Lcannot be aθ∗-completion ofH.By the defnition ofθ∗-completions,Lcontains properly a normal subgroup ofG.In particular,N≤L,which yieldsG=LN=L,a contradiction.We hence follow thatCmust be a maximal subgroup ofG.

Lemma 2.7A subgroupHofGisF-abnormal inGif and only i

ProofIfG/HG∈F,thenGF⊆HG⊆H.Conversely,ifGF⊆H,thenSoG/HG～=(G/GF)/(HG/GF)∈F.We hence have thatHisF-normal inG.

Lemma 2.8[8,Theorem3]For any fnite groupG,the intersection of maximal subgroups ofGof composite index is supersolvable.

Lemma 2.9Φ1(G)is solvable.

ProofIfF(G)is an empty set,that is,the groupGhas not 2-maximal subgroups,it follows thatGis of prime order.SoGis supersolvable,we hence conclude that Φ1(G)=Gis solvable,a contradiction.ThusF(G)/=Øand Φ1(G)/=1,then it follows thatGis not simple.LetNbe a minimal normal subgroup ofG,the minimality ofGimplies that Φ1(G/N) is solvable.On the other hand,Φ1(G)N/N≤Φ1(G/N),we have that Φ1(G)N/Nis solvable.

IfN∩Φ1(G)=1,then Φ1(G)is solvable,a contradiction.So we have thatN∩Φ1(G)/=1,that is,N≤Φ1(G)and Φ1(G)/Nis solvable.Moreover,Nis the unique minimal normal subgroup ofG.

IfN≤L(G),whereL(G)is the intersection of maximal subgroups ofGof composite index, thenNis solvable,which is absurd.So there exists a maximal subgroupLof composite index such thatN/≤L.ForthenN/≤A.From|G:L|||G:A|,then|G:A|is a composite.It concludes thatN≤Φ1(G)≤A,which is a fnal contradiction.

Lemma 2.10[9,Lemma2.16]LetFbe a saturated formation containingUandGbe a group with a normal subgroupEsuch thatG/E∈F.IfEis cyclic,thenG∈F.

Lemma 2.11[4,Lemma1.1]A maximal subgroupMofGisF-abnormal inGif and only ifG=GFM.

Lemma 2.12[4,Lemma1.6]LetFbe a saturated formation containingU.ThenG∈Fif and only if everyF-abnormal maximal subgroup ofGhas index a prime.

Lemma 2.13[10,Lemma3]LetMbe a subgroup ofG.IfMhas square-free order and prime index,thenGis solvable.

Lemma 2.14[10,Lemma2]If a groupGis factorizable in the formG=CD,where neither|C|nor|D|is divisible by 4,thenGis 2-nilpotent,in particular,Gis solvable.

§3.Main Results

Theorem 3.1LetFbe a saturated formation which containsU.For everyH∈Fc(G),Hhas a maximalθ∗-completionCsuch thatG=CHandC/HGhas square free order,thenG∈F.

ProofAssume thatGdoes not belong toF.We work for a contradiction.By Lemma 2.2,there exists a normal subgroupNofGsuch thatG/N∈b(F).ThenG/Nhas a unique minimal normal subgroupU/N.We hence have thatU/N=(G/N)F=GFN/N.

(1)There exists anM∈Uc(G)such thatN≤M.

IfUc(G/N)=Ø,thenG/N∈Fby Lemma 2.12,contrary toG/N/∈F.SoUc(G/N)/=Ø, that is,there exists a maximal subgroupMofGof composite index such thatN≤Mwhile.In particular,.By Lemma 2.11,we haveM∈Uc(G).

(2)U/Nis a non-abelian characteristic simple group.

TakeMas in(1).For any,thenHhas composite index inGandBy the hypothesis,Hhas a maximalθ∗-completionCsuch thatG=CHandhas square free order.Assume thatU/Nis solvable,then|U/N|is a power of a primep,andis a power of a primep.That is,Mhas prime power index.Fromwe have thatmust be square free.Becausemust be square free.ThusMhas prime index inG,a contradiction.

(3)N=HG,Cis maximal inCUand

TakeHas in(2).That is,Hhas composite index inGand.Since,it follows thatN=HGby Lemma 2.3.BecauseU/Nis non-solvable,whileC/Nis of square-free order and hence solvable,we have thatWe thus can apply Lemma 2.4 to see thatCis maximal inCU.I,thenSo we have thatU/Nis solvable, which is absurd.

(4)Cis maximal inG.

By Lemma 2.6,Cis maximal inG.

(5)Final contradiction.

If|G:C|is a prime,thenC/Nhas prime index inG/N.AsC/Nis square free order,G/Nwould be solvable by Lemma 2.13,a contradiction.SoChas composite index inG.Thus, forit is clear thatApplying the hypothesis,we see thatLhas a maximalθ∗-completionDsuch thatG=LDandD/LGhas square free order.Sinceit concludes thatLG=Nby Lemma 2.3.Now,G/N=L/N·D/Nand bothL/NandD/Nhave square free order.By Lemma 2.14,G/Nwould be solvable,which is afnal contradiction.

Theorem 3.2LetFbe a saturated formation which containsU.Every memberHofFc(G)has a maximalθ∗-completionCsuch thatCG/HGis cyclic,thenGbelongs toF.

ProofAssume thatGdoes not belong toF.LetGbe a counterexample.Then there exists a normal subgroupNofGsuch thatG/N/∈FandU/Nis the unique minimal normal subgroup ofG/N.Moreover,U/Nis non-cyclic and we haveU/N=(G/N)F=GFN/N.

(1)There exists anM∈Uc(G)such that

As in the proof of Theorem 3.1,we have

(2)Final contradiction.

TakeMas in(1).For anythen|G:H|is a composite andBy the hypothesis,Hhas a maximalθ∗-completionCsuch thatCG/HGis cyclic.Sinceit follows thatN=HGby Lemma 2.3.The uniqueness ofU/Nimplies thatSo we have thatU/Nis cyclic,a contradiction.

[1]ZHAO Yao-qing.On the Deskins completions,theta completions and theta pairs for maximal subgroups[J]. Comm Algebra,1998,26:3141-3153.

[2]ZHAO Yao-qing.On the Deskins completions,theta completions and theta pairs for maximal subgroups[J]. Comm Algebra,1998,26:3155-3164.

[3]WEI Hua-quan.Onc-normal maximal and minimal subgroups of Sylow subgroups of fnite groups[J].Comm Algebra,2001,29:2193-2200.

[4]LI Shi-rong.OnF-abnormal maximal subgroups of fnite groups[J].Acta Math,2007,23:885-888.

[5]GUO Wen-bin.The Thoery of Classes of Groups[M].Beijing,New York,Dordrecht Boston,London:Science Press Kluwer Academic Publishers,2000.

[6]LI Shi-rong,ZHAO Yao-qing.Ons-completions of maximal subgroups of fnite groups[J].Algebra Colloq, 2004,11:411-420.

[7]XU Ming-yao,HUANG Jian-hua,LI Hui-ling,et al.Introduction to Finite Group[M].Beijing:Science Press,1999.

[8]BHATTACHARYA P,MUKHERJEE N P.On the intersection of a class of maximal subgroups of a fnite group[J].J Pure Appl Algebra,1986,42:117-124.

[9]SKIBA A N.On weaklys-permutable subgroups of fnite groups[J].J Algebra,2007,315:192-209.

[10]LI Shi-rong.The Deskins index complex and the supersolvability of fnite groups[J].J Pure Appl Algebra, 1999,144:297-302.

[11]GAO Hui,GAO Sheng-zhe,YIN Li.Ons∗-completions of maximal subgroups of fnite groups[J].Chin Quart J of Math,2015,30(3):381-388.

[12]LI Chang-wen,HU Bin.A note ons-quasinormally embedded andc-supplemented subgroups of fnite groups[J].Chin Quart J of Math,2012,27(2):213-217.

[13]WU Ming-zhong.The derivation algebra ofQnfliform Lie algebra[J].Chin Quart J of Math,2013,28(3): 397-401.

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1002–0462(2016)04–0406–06

Received date:2014-10-23

Biographies:GAO Hui(corresponding author)(1978-),female,native of Zhuanghe,Liaoning,a lecturer of Dalian Ocean University,engages in fnite group;GAO Sheng-zhe(1974-),male,native of Daxinganling, Heilongjiang,an associate professor of Dalian Ocean University,engages in fnite group;YIN Li(1981-),female, native of Zhuanghe,Liaoning,a lecturer of Dalian Ocean University,engages in fnite group.

2000 MR Subject Classifcation:20D10