该【on some closure properties of the non-abelian tensor product 2017 g. donadze参考 】是由【小舍儿】上传分享,文档一共【19】页,该文档可以免费在线阅读,需要了解更多关于【on some closure properties of the non-abelian tensor product 2017 g. donadze参考 】的内容,可以使用淘豆网的站内搜索功能,选择自己适合的文档,以下文字是截取该文章内的部分文字,如需要获得完整电子版,请下载此文档到您的设备,方便您编辑和打印。:..AcceptedManuscriptOnsomeclosurepropertiesofthenon-,,:S0021-8693(16)30446-XDOI:http://dx./.:YJABR15998Toappearin:JournalofAlgebraReceiveddate:28December2015Pleasecitethisarticleinpressas:.,Onsomeclosurepropertiesofthenon-abeliantensorproduct,(2017),http://dx./.,typesetting,andreviewoftheresultingproofbeforeitispublishedinits?,andalllegaldisclaimersthatapplytothejournalpertain.:..Onsomeclosurepropertiesofthenon-,,?,,Thiruvananthapuram,IndiabDepartmentofAlgebra,postela,15782SantiagodeCompostela,SpainAbstractWeprovethattheclassofnilpotentby?nite,solvableby?nite,polycyclicby?nite,nilpotentofnilpotencyclassnandsupersolvablegroupsareclosedundertheformationofthenon--saryandsu?cientconditionsforthenon-abeliantensorproductof?nitelygeneratedgroupstobe?:Schurmultiplier,non-abeliantensorproduct2010MSC:20D99,20F16,20F05,20F80,20G05,?nitenesspropertiesofthenon-abeliantensorproductG?.--abeliantensorproductG?HforapairofgroupsGandHin[5]and[6]inthecontextofanapplicationinhomotopytheory,[15].Wewerenaturallyledtothestudyoftheclosurepropertiesofnon-abeliantensorproductofgroupswhileconsideringthequestionwhethertheSchurmultiplierofNoetheriangroupsis?[7]and[12],theauthorsprovethatthenon-abeliantensorproductof?nitegroupsisa?nitegroup,andtheyalsoshowthatthenon-abeliantensorproductof?nitep-groupsisa?nitep-[13],Visscherprovedthat?:+34881813138,Fax:+:gdonad@(),manuel.******@(),******@()PreprintsubmittedtoJournalofAlgebraDecember1,2016:..ifG,Haresolvable(nilpotent),thenG?Hissolvable(nilpotent).In[10],NakaokaalsoprovedthatifGandHaresolvable,thenG?-[13]givesaboundonthenilpotencyclassofG?HintermsofthederivativesubgroupDH(G)-abeliantensorproductofgroupsofnilpotencyclassatmostnisagroupofnilpotencyclassatmostn,therebyimprovingtheboundgivenbyVisscherin[13].Asacorollary,weobtainaboundonthenilpotencyclassofG?Gwhichisanimprovementoftheboundobtainedbytheauthorsof[1].In[9],MoravecprovedthatifGandHarepolycyclicgroups,thenG?-,,wegiveshortproofsofthemainresultsin[13].WealsoprovethatifGandHaresupersolvablegroups,thenG?[2]provethatthenon-abeliantensorsquareofnilpotentby?nitegroupisanilpotentby?-abeliantensorproductofnilpotentby?nitegroupsisanilpotentby?-abeliantensorproductofsolvableby?nitegroupsissolvableby?,weprovethatthenon-abeliantensorproductoflocally?nite,locallysolvable,locallynilpotent,,weprovethe?nitenessofG?,weconsidertheclassofgroupsGwhichisanextensionofa?nitelygeneratednon-abelianfreegroupbya?nitegrouporitisanextensionofa?nitegroupbya?nitelygeneratednon-,weprovethatifHisa?nitegroup,thenG?Hisa?,weprovethatifGisa?nitelygeneratedgroupandHisapatibly,withtheactionofHonGbeingtrivial,thenG?Hisa?,weaddressthefollowingquestion:isthenon-abeliantensorproductof?nitelygeneratedgroups?nitelygenerated?Ingeneralthisneednotbethecase.?Htobe?,thenwegivenecessary2:..??nitegroups,thenG?Hisapolycyclicby?nitegroup,-abeliantensorproductofgroupsisde?nedforapairofgroupspatibilitycondi-tionsofDe?,sogg=ggg?,g∈Gandgg·g?1=[g,g]mutatorofgandg.De?-hgh?1ghghg?1h=handg=g,forallg,g∈G,h,h∈?,thenthenon-abeliantensorproductG?Histhegroupgeneratedbythesymbolsg?∈Gandh∈Hwithrelationsgg?h=(gg?gh)(g?h),g?hh=(g?h)(hg?hh),forallg,g∈Gandh,h∈=H,andallactionsaregivenbyconjugation,iscalledthetensorsquareG??:G?G→[G,G]sendingg?hto[g,h].SetJ(G)=Ker(κ).ItstopologicalinterestistheformulaJ(G)~=π3(SK(G,1)),whereSK(G,1)isthesuspensionofK(G,1).ThegroupJ(G)liesinthecentreofG??derivativeofGbyHwasintro-ducedin[13].Itisde?nedasDH(G)=ghg?1|g∈G,h∈-knownconceptofacrossedmodulecanbefoundin[6].In[14],:..De?:A→BtogetherwithanactionofBonAsatisfyingφ(ba)=bφ(a)b?1andφ(a)a=aaa?1,forallb∈Banda,a∈[6,].:G?H→D(G)bede?nedbyφ(g?h)=ghg?:(i)φisahomomorphism;(ii)thereisanactionofGonG?Hde?nedbyx(g?h)=xg?xh,wherex∈G;(iii)φ:G?H→DH(G):A→-abeliantensorproductofgroupsIfGandHbelongtoclassX,thendoesG?HbelongtoclassX?[7],[13],[10]and[9]haveconsideredthisquestionwhenXistheclassof?nitegroups,p-groups,solvablegroups,?-?nite,solvableby?nite,locally?nite,locallynilpotent,locallysolvable,locallypolycyclicandlocallysupersolvablegroupsareclosedundertheformationofthenon-→A→G?H?→DH(G)→,(G)issolvableor4:..anilpotentgroup,thenG?HbeingacentralextensionofDH(G)[13],thenthefollowinghold:(i)IfDH(G)isabelian,thenG?Hismetabelian.(ii)IfDH(G)issolvable,thenG?Hissolvable.(iii)IfDH(G)isnilpotent,thenG?[13]and[10]showthatifGandHarenilpotentgroupsofclassn,thencl(G?H)≤cl(DH(G))+1,(G)isn,inwhichcasetheaboveformulagivesanupperboundofn+,weimprovethisboundanditprovidesanotherexampleoftheclosurepropertyofthenon-,thenG?(n+1)-thtermofthelowercentralseriesγn+1(G?H)?1=yforeachx∈γ(G?H)nandy∈G??cestoshowthatconjugatingg?hby[...[[g1?h1,g2?h2],g3?h3],...,gn?hn]?xesg?hforeachg,g1,...,gn∈Gandh,h1,...,hn∈[4,Proposition3],(a?b)(a?b):=(a?b)(a?b)(a?b)?1=[a,b](a?b)=[a,b]a?[a,b]?bisthesameasactionby[a,b].patibilityoftheactions,weobtain[...[[g1?h1,g2?h2],g3?h3],...,gn?hn](g?h)=[...[[[g1,h1],[g2,h2]],[g3,h3]],...,[gn,hn]](g?h)=[...[[[g1,h1],[g2,h2]],[g3,h3]],...,[gn,hn]]g?[...[[[g1,h1],[g2,h2]],[g3,h3]],...,[gn,hn]]h[...[[gh1g?1,gh2g?1],gh3g?1],...,gnhngn?1][...[[g1h1h?1,g2h2h?1],g3h3h?1],...,gnhnh?1n?1n]=112233g?123h=g?:..IfGisanilpotentgroupofnilpotencyclassn,thenby[1,]cl(G?G)=cl([G,G])orcl([G,G])+,thenclearlythenilpotencyclassof[G,G][1],weobtainthatthecl(G?G)≤n+[4],[13]and[10].Sincetheproofissimilartotheproofofthepreviousproposition,?Gisboundedabovebyn,wherex-?Hasolvablegroupoflengthatmostn?Wedonotknowtheanswertotheabovequestionevenforthecasen=2,,weprovethatthepropertyofbeingsupersolvableisclosedunderformationofnon-,thenG?:φG1→A→G?H?→DH(G)→1,whereφ:g?h →ghg?1foreachg∈G,h∈?H→D(G)GHisacrossedmodule,AisasubgroupofthecenterofG?,weconcludethatG?Hispolycyclic[9].HenceAis?nitelygeneratedandisisomorphictothedirectproductnof?nitelymanycyclicgroups,A=⊕=1ofG?H,itisanormalsubgroupofG?Hforeach1≤i≤,1→nnA1→(G?H)/⊕Ai→(G?H)/A=DH(G)→(G?H)/⊕Aii=2i=,1→A2→(G?6:..nnH)/⊕Ai→(G?H)/⊕Ai→=3i=2ngroupbysupersolvablegroupimplyingsupersolvabilityof(G?H)/⊕=3Proceedingbyinduction,wewillobtainthatG?,wewanttoexaminewhetherG?-posePisapropertyofgroupsthatsatis?esthefollowingconditions:(i)Pisclosedundertakingnormalsubgroups;(ii)IfagrouphaspropertyP,?→φGφHKerφG→G?H?→DH(G)→1or1→KerφH→G?H?→DG(H)→?niteornilpotentby?niteisclosedundertakingnormalsubgroupsandalsoclosedundertakingcentralextensions,,thenthefollowinghold:(i)IfGorHissolvableby?nite,thenG?Hissolvableby?nite;(iii)IfGorHisnilpotentby?nite,thenG?Hisnilpotentby?,weonlyrequireDG(H)orDH(G)?nite,solvable,supersolvable,nilpotentorpolycyclicgroups,thenG?Halsobelongstothesameclass,-abeliantensorproductofgroupsifGandH7:..havepropertyPimpliesthatG?Hh
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