Article信州大学教養部紀要. 第二部, 自然科学 8: 1-4(1974)departmental bulletin pape

NISHIKAWA,  Akira

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Shinshu University Institutional Repository

A Note on the Dimension Subgroups Akira NISHIKAWA     §  I. Introduction    The n th dimension subgroup  Dn(G) of a group G is the subgroup of G consist-ing of elements x such that  x-1E/nG+1, where  /G is the augmentation ideal of the group ring ZG.    On the other hand, we write  Gn as the n th term in the lower central series of G.    The dimension conjecture is referred to  :  Dn(G)=Gn for all groups G and all integers  n>0.    The case  n=0 is trivial, and the case n=1 is proved from the fact  G/G1=  //J.', etc. (1).    This conjecture was introduced by MAGNUS.    He established when G is any free group. (5).    Attempts by CORN and LOSEY to establish this conjecture were unsuccessful except in a special case (6).    HIGMANN proved that to establish the dimension conjecture it is enough to give an affirmative answer when G is any finite p-group (1), and from the process of his proof, the following important theorem was induced (1,  6)  :    THEOREM 1. If G is a group such that  Dn(G)7L-Gn, then there is a subquotient group L of G with the properties  :     (a) L is a finite  P-group of  classn.     (b)  Dn(L),-L{    In this paper the auther tries to give another proof of this theorem, which is successful in a very special case.    REMARKS (1)  Dn(G)DGn is an easy exercise.    (2) PASSI proved the following results using THEOREM 1. and various coho-mological methods (6)     (a) D2(G)=G2 for all groups G.     (b) If  G is a group such that none of its subquotients is a 2-group of class 3, then  Ds(G)=  G3.  (c) If  G is a group such that all its subquotients which are 2-groups of class 3 have order 64, then  D3(G)=G3.    (3) CORN and LOSEY proved  Dn(G)=Gn, for all  n>0, when  G  satisfies  :  xEG„,  xE  Gn+,  xk  EGn+ixk  =1.2　　　　　　g2．　itiffifaegEsioit　seshgroups　ef　a　qesetaestt　group　　　Let　G　be　a　group，　T＝＝Q／Z　the　additive　group　of　rationals　mod　1，　and　g　amapping　G．T　of　the　underlying　set．　　　The　mapping　q　can　be　extended　by　liniarity　to　a　homomorphism　of　the　add－itive　group　of　the　group　ring　ZG　to　T．　　　g　：　G－T　is　called　a　polynomlal　map　of　degree＝〈n　if　p　vanishis　on　1”’i．　　　　　LEMMA　1．　lf　G　is　a　group，　thenDn（G）＝＝｛：x　i　x∈0，　andψ（x）＝＝g（1）for　all　polynomia1皿aps　g：G→7「of　degree≦n＝｝（6）．　　　　　LEMMA　2．　lf　G　is　a　group，　and　N　a　normal　subgroup　of　G，　then／面差（塔＋1＋（■G）ムう／（ZG）IN｛：IIItt＋1／（ZG）緬塔＋1．　　　If　1＞is　contained　in∠）n（G），　then／3力ξ主1懲＋1／（ZG）1撞．　　　PRooF．　Let　it：　ZG．Z（G／N）　be　the　ring　epimorphism　induced　from　the　nat－ural　group　epimorphism　T　：　G．G／N，　then　Ker　it　is　obviously　（ZG）JN，　the　ideal　ofZG　g・n・・at・d　by　th・augm・nt・ti・n　id・al・f　ZN．元m・p・1警＋1・nt・塔於，・・th・upper　isomorphisms　are　established．　　　Now　let　N　be　contained　in　Dn（G），　then　｛g（h－1）｝，　where　gEG　and　h　Eii　Ai，　gen－erates　（ZG）IN　over　Z．　ebserving　g（h－1）EI”G“i，　we　can　see　（ZG）IAr　c　l”G＋i，　whichleads　to　the　lower　isomorphism．　Therefore　the　lemma　is　proved．　　　PRoPoslT工oN　1．亙f　2＞is　a　normal　subgroup　of　a　group　O，　thenDn（G／N）　＝Dn（G）N／N．　　　Especially　if　N　is　contained　in　Dn（G），　then　Dn（G／N）　Dn（G）／N．　　　PRooF．・　Let　gNEDn（G）N／N，　¢：　G／N－T　be　any　polyno．mial　map　of　degree；＄；n，　then　we　can　assume　that　g　belongs　to　Dn（G）．　Now　g＝ipz，　where　T　is　thenatural　epimorphism　of　G　onto　G／IV，　is　a　polynomial　map　G．T　of　degreegnbecause　9（勢＋1）⊂ψ（塔方｝）＝0．　So　from　9（≡1）n（G）and　LEMMA　1，　it　followsψ（9：N）＝P（．cr）＝g（1）＝＝ip（N）．　Therefore　gArED．（G／N）．　　　Conversely　let　gNEiDn（G／N）．　Then　gN－NEiilC’7“，　and　observing　the　upper　iso－morphisms　of　LEMMA　2，　we　can　seeg一1十（ZG）ILv　E（1’Gt“’十（ZG）11v）／（ZG）IN，　which　leads　to　gh－I　E　I’G’＋i，　where　h　is　a’suitableelement　in　N．　Thereforeψ6D72（G），　i．　e．幽V∈1）n（G）ム1／N．　　　Now　the　first　equality　is　established．　　　The　second　equality　is　naturally　induced　from　the　first．　　　CoRoLLARy　1．　lf　G　is　a　group　such　that　Dn（G）；Gn，　then　there　is　a　quotientgroup　K　of　G　with　the　properties　：　　　　　（a）　K　is　a　nilpotent　group　of　class．〈＝．n．　　　　　（b）　Dn（K）“ff｛1｝　　　PRooF．　’Put　K＝G／Gn，　theit　clearly　K　is　a　nilpotent　group　of　class；Sn，　andby　the　proposition，A　Note　on　the　Dimension　Subgroups 3Dn（K）　＝Dn（G）／Gn　￥｛　1　｝　　　This　completes　the　proof．　　　　　CoRo－ARy　2．　ln　the　hypothesis　of　the　proposition，　there　iS　a　one－to－onecorrespondence　between　the　set　of　all　polynomial　maps　G－T　of　degree；Sn　andthe　set　of　all　polynomial　maps　G／N．T　of　degree；：Sln．　　　Especially　there　is　a　one－to－one　correspondence　between　the　set　of　all　polyn－ornial　maps　G－T　of　degree＄n　and　the　set　of　all　polynomial　maps　G／Dn（G）一T　ofdegreeSn．　　　　　PRooF．　For　a　polynomial　map　¢　：　G／N．　T　of　degreel：Eln，　we　correspond　¢r　：G－G／N－T　which，　as　was　seen　in　the　proof　of　the　proposition，　is　a　polynomjalmap　of　degreeiSn．　　　Conversely　let　q　：　G－T　be　a　polynomial　map　of　degreelg＄n．　　　Then　observing　h－I　E　I’G’＋i，　we　see　：　　　g（gh）一g（g）一g（gh－g）＝g（g（h－1））Eg（f’．t’i）＝o．　　　So　the　image　of　gN　by　q　is　uniquely　deterrniRed　and　the　xnap　¢：　G／N－T　isinduced　such　that　g　＝＝¢n　　　The　fact　that　if　is　a　polynomial　map　of　degree．〈，．n　is　easily　seen　by¢（ノ勘わ一φ・（∫各＋1）一9（∫警＋1）一〇。　　　The　fact　that　those　correspondences　are　inverse　to　each　other　is　trivial．　　　This　completes　the　proof．　　　　　　g3．　A　proof　of　WHEoREM　g　in　ehe　case　G／Gn　is　fiptte　　　LEMMA　3．　gf　G，　ff，　．　．　．　，　K　are　groups，　then　　　　　　　（G×ff×．．．　×K）n　＝Gn　X　Hn×．．．　×Kn，　　　　　　　Dn（G×HX．．．　×K）　＝Dn（G）×Dn（ff）×．．．　×Dn（K）．　　　PRooF．　The　upper　equaltiy　is　suitable　as　an　exercise　in　Greup　Theory，　andwe　omit　its　proof．　　　Let　f：　Z（G　x　ff）一ZG　be　the　ring　epimorphism　induced　from　the　projectionGxH－G．　Then　easily　f（PG”．iff）＝fG．　　　LeteGDn（G　x　H）．　We　write　a＝＝（g，　h），　gGG，　hEff．　　　　　From　the　fact　a－IEI’G”．iff，　it　follows　g－1＝：f（es－1）Ei　l’GZ“i，　so　gG　D．（G）．　Si－milarly　hEii　Dn（H）．　　　Now　we　have　Dn（G　x／f）aDn（G）xDn（if），　and　the　converse　is　trivial．　　　Coitsequently　it　follows　：　　　　　　　Dn（G　×HX　．　．　．　×K）　＝＝　Dn（G）　×　Dn（ff×．　．　．　×K）　＝．　．　．　＝＝　Dn（G）　×　Dn（H）　×．　．　．　×　Dn（K）．　　　This　completes　the　proo£　　　　　CoRoLLARy．　lf　there　is　a　finite　nilpotent　greup　G　of　class；：iln　such　thatエ）n（G）≒｛二1｝，　then　there　is　a　finite　P－group　Z．　of　class≦n　such　that　Dn（L）≒｛1｝．4　　　　　PRooF．　As　any　fiRite　nilpotent　group　is　the　direct　product　of　its　Sylowsubgroups，　we　find　a　suitable　Sylow　subgroup　of　G　as　L　in　the　corollary．　　　This　completes　the　proof．　　　　　Combining　COROLLARY　l　to　PROpOslTION　1　and　COROLLARy　to　LEMMA　3，THEoREM　1　is　verified　iR　the　case　G／Gn　is　finite．鼠efe彊『e1瓶ces1．　BABAKHANiAN，　A．　：　Cohomologica1　Methods　in　Group　Theory；　Marcel　Dekker，　lnc．　（1972）．2．　CoRN，　P．　M．：Generalization　ef　a　theorem　of　Magnus；Proc．　London　Math．　Soc．　57　（1952）．3．　LosEy，　G．　：On　dimension　subgroups；Trans．　AMS．　97　（1960）．4．　LosEy，　G．：Math．　Reviews　26，　6260，　p．　1187　（1963）．5．　MAGNus，　W．　：　Uber　Beziehungen　zwischen　h6heren　Kommutatoren　；　」．　reine　angev．　Math．　177　（1937）．6．　PAssi，　1．　B．　S．：Dimension　Subgroups．；J．　Alg．　9（1968）．SUPP亘e簸亙e四重ary　Nete　　　The　auther　lectured　on　the　contents　of　this　paper　in　the　18　th．3，　1974　at　thesocietyプ「or　the　stndy　oゾCohomology　Theory　of　2宥吻∫6　（⊇roups　in　The　工nstitute　ofMathematical　Analysis，　Kyoto．

A Note on the Dimension Subgroups

NISHIKAWA  Akira

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A Note on the Dimension Subgroups

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