F-S-可補(bǔ)的子群對(duì)有限群結(jié)構(gòu)的影響

趙勇

(廣安職業(yè)技術(shù)學(xué)院教育系,四川 廣安 638000)

F-S-可補(bǔ)的子群對(duì)有限群結(jié)構(gòu)的影響

趙勇

(廣安職業(yè)技術(shù)學(xué)院教育系,四川 廣安 638000)

設(shè)F是一個(gè)群系.群G的一個(gè)子群H在G中F-S-可補(bǔ),如果存在G的子群K,使得G=HK且K/K∩HG∈F,其中HG表示G包含在H中的最大的正規(guī)子群.本文利用群系理論研究子群的F-S-可補(bǔ)性對(duì)有限群結(jié)構(gòu)的影響,得到如下結(jié)論:設(shè)F是子群閉的局部群系,G是有限群且GF是可解的.則G∈F的充要條件是下列條件之一:(1)G存在正規(guī)子群N使得G/N∈F且N的極小子群及4階循環(huán)子群(p=2)均在G中F-S-可補(bǔ).(2)G存在正規(guī)子群N使得G/N∈F,N的4階循環(huán)子群在G中有F-S-補(bǔ)且N的極小子群皆包含在ZF∞(G)中.應(yīng)用這些結(jié)論,可以得到一些推論,其中包括已知的相關(guān)結(jié)果.

群系;F-S-可補(bǔ);極小子群

1 引言

1996年,文獻(xiàn)[1]引進(jìn)了C-正規(guī)子群的概念,并證明:如果有限群G的極小子群及4階循環(huán)群在G中C-正規(guī),則G是超可解群.1999年,文獻(xiàn)[2]證明了G的所有素?cái)?shù)階子群在G中都有補(bǔ)的有限群恰好是Sylow子群是初等交換群的超可解群.還有許多學(xué)者利用極小子群的特殊性質(zhì)研究有限群的結(jié)構(gòu),文獻(xiàn)[3]證明了:若G的每個(gè)素?cái)?shù)冪階循環(huán)子群是G的弱s-補(bǔ)子群,則G為超可解群.近來,文獻(xiàn)[4]引入了F-S-補(bǔ)的概念,并利用了某些子群的F-S-可補(bǔ)性給出了群結(jié)構(gòu)的新刻畫.本文將在以上的結(jié)論的基礎(chǔ)上運(yùn)用群系理論來討論極小子群在F-S-補(bǔ)的條件下對(duì)有限群結(jié)構(gòu)的影響.

一個(gè)群類F稱為群系,如果它關(guān)于同態(tài)像和次直積都是封閉的.一個(gè)函數(shù)f稱為一個(gè)群系函數(shù),如果對(duì)于任意素?cái)?shù)p,f(p)為一個(gè)群系.一個(gè)群系F稱為局部的,如果存在一個(gè)群系函數(shù)f滿足F={G|G/CG(H/K)∈f(p),對(duì)于G的所有主因子H/K且p||H/K|},此時(shí)稱f局部定義了群系F,并記作F=LF(f).如果一個(gè)群系滿足條件:由G/Φ(G)∈F總有G∈F,則稱F為飽和群系.熟知,一個(gè)群系是局部的當(dāng)且僅當(dāng)它是飽和的.在本文中U表示所有超可解群構(gòu)成的群類,Np表示所有p-冪零群構(gòu)成的群類,N 表示所有冪零群構(gòu)成的群類.明顯地,U,Np,N都是子群閉的局部群系.文中所有群為有限群.未交待的定義和符號(hào)是標(biāo)準(zhǔn)的.

2 預(yù)備知識(shí)及引理

3 主要定理

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[2]Ballester-Bolinches,Guo X Y.On complemented subgroup of fi nite groups[J].Arch.Math.,1999,72(3):161-166.

[3]郭鵬飛,郭秀云.弱s-置換性傳遞的有限群[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2009,25(4):649-653.

[4]Miao L,Guo W.Finite groups with some primary subgroupsF-S-supplemented[J].Comm.Algebra, 2005,33(8):2789-2800.

[5]Guo W.The Theorey of Classes of Groups[M].New York:Science Press,2000.

[6]Li Shirong.On minimal subgroups of fi nite groups III[J].Communications in Algebra,1998,26(8):2453-2461.

[7]徐明曜.有限群導(dǎo)引:上[M].北京:科學(xué)出版社,1993.

[8]李長穩(wěn),郭文彬.關(guān)于F-S-可補(bǔ)子群[J].數(shù)學(xué)研究與評(píng)論,2007,27(1):207-211.

The in fl uence of F-S-supplemented subgroups on the structure of fi nite groups

Zhao yong
(Department of Education,Guang′an Vacational and Technical College,Guang′an 638000,China)

LetFbe a class of groups.A subgroupHis calledF-S-supplemented inGif there exists a subgroupKofGsuch thatG=HKandK/K∩HG∈F,whereHGis the maximal normal subgroup ofGcontained inH.In this paper,theF-S-supplemented subgroups ofGis used to study the structure ofGby the theory of formations.The following results are obtained:LetGbe a fi nite group,Fbe a formation andGFbe soluble. ThenG∈Fif and only if one of the following two conditions:(1)Ghas a normal subgroupNsuch thatG/N∈Fand the subgroups of orderpor 4 ofNareF-S-supplemented inG.(2)Ghas a normal subgroupNsuch thatG/N∈F,the subgroups of prime order ofNare contained in(G)and the subgroups of order 4 areF-S-supplemented inG.By these results,we may get a series of corollaries,which contain known results.

formations,F-S-supplemented,the minimal subgroups

O152.1

A

1008-5513(2012)05-0614-06

2012-01-16.

四川省學(xué)術(shù)委員會(huì)基金(SZD0406).

趙勇(1982-),碩士,講師,研究方向:有限群論.

2010 MSC:20D10,20D15,20D20