On solvable groups, all proper factor groups of which have finite ranks
This paper deals with finitely generated finitely approximable solvable groups of infinite special rank, all proper normal subgroups of which determine the factor groups of finite special ranks.
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| Datum: | 1993 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Russisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
1993
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/5930 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512152968757248 |
|---|---|
| author | Tushev, A. V. Тушев, А. В. Тушев, А. В. |
| author_facet | Tushev, A. V. Тушев, А. В. Тушев, А. В. |
| author_sort | Tushev, A. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-19T09:21:13Z |
| description | This paper deals with finitely generated finitely approximable solvable groups of infinite special rank, all proper normal subgroups of which determine the factor groups of finite special ranks. |
| first_indexed | 2026-03-24T03:24:15Z |
| format | Article |
| fulltext |
Y,UK512.544
A. B. Tyweo, KaH11. q:>H3.-MaT. HayK (U11enpone-rp. yH-T)
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We consider finitely generated residually finite soluble groups of infinite rank, all of whose proper
quotient groups have a finite rank.
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1274 ISSN 0041-6053. YKp. Mam. )l(YP"·· 1993, m . 45 , N• 9
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= aKH, fl= Ste([) H / = Annxia).
ISSN 0041 -6053. Y,:p. ,,,a11z. ;;cypn. , 1993, m. 45, N• 9
1278 A. B. TYllIEB
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HMeeT 6eCKOHe'IHbl0 paHr. CneJ].OBaTeJihHO, nOJ].MOJJ.YJib U Kr onpeaen.lleT
q>aKTOp-rpynny M / UKr 6ecKoHe'IHOro paHra, 'ITO Heso3MO.>KHO. TaKHM o6pa3oM,
V - rrpocTo0 KH-MOJJ.YJib, a TaK KaK V - cpnHHTHO-annpoKCHMHpyeMhIA MOJJ.YJib,
TO IVl< 00,H, CJieJ].0BaTMbHO,IH/CH<V)l< 00 • TIOCKOJlbKY A$H, TO An$CH<V),
a TaK KaK An - HOpMaJibHa.ll noarpynna rpynnb1 r. TO A" $ Cr(M '). Toraa,
O'leBHJJ.HO,r 0(F / Cr(M ')) $ r0( r).
Tipeanono.>KHM Teneph, 'ITO Ir: HI< oo, H nycTb P - npocT00 HJJ.eaJI H3 KA,
J].Jl.ll KOTOporo P n KD =PO= I n K.D. Tor JI.a no JieMMe 4 H =· Str(P ). TiyCTb K
- none 'laCTHhlX KOJibQa K u P - uaean s KA, nopO.>KJJ.eHHbIA P. Tor aa
tterpyJJ.HO noKa3aTb, 'ITO H = Str(P), a TaK KaK Ir: HI< 00, TO no TeopeMe A H3
pa6oThl [5] j,+ = A n (P + 1) "# 1. TaK KaK MOJJ.YJib M 6e3 K-Kpy'leHH.ll, TO
P n K = 0, "· CJieJ].OBaTeJibHO, j,+ = p+. 0'-leBHJJ.HO, D n P + $ CA (a) = C.
TiocKOJibKY MOJJ.YJib M cpmrnTtto-annpoKCHMupyeMhitt, To KA-MOJJ.YJib aKA TaK.>Ke
q>HHHTHO-annpOKCHMHpyeM, H, CJieJ].OBaTeJibHO, <PHHHTHO-annpOKCHMHpyeM00
6yJJ.eT q>aKTOp-rpynna A IC. 0TCIOJJ.a BBHJJ.Y MHHHMaKCHOCTH A It~ IC) I < oo, u,
3aMeHHB A ee r-aonyCTHMOA noarpynnott KOHe'IHOro HHJ].eKca, MO.>KHO C'IHTaTb,
'ITO noarpynna C cepsaHTHa s A. TaK KaK H = StrU) H C =A n ( 1 + M ), TO
JJ.Jl.ll , 1!06oro g EH cymecrnyeT KOHe'IHonopo.>KJJ.eHHa.ll noarpynna Ag $A TaKM,
'ITO r(Ag) = r(A) H c g n Ag= C n A g =Cg· Tioarpynna C8 cepBaIITHa B A,
n03TOMY C8 /Cg = t ( A /Cg) = C /Cg• "· CJieJ].OBaTeJibHO, C8 = C. TaKHM o6pa30M, C -
HOpMa1IbHM noJI.rpynna B H, oTKy aa A $ CH< V ), H, CJieAOBaTeJrbHO, r(H IC H<V)) <
< r(r). TeopeMa JJ.OKa3aHa.
JleMMa 5. flycmb S - KOMMymam1umoe KOll bl(O, r - zpynna ew onepa
mopoe, M - S-Mooy11b, 0 "# a E S, u F -nwKOu ceo6oOHblu no0Mooy11b e M, limo
11106ou 3/leMeHm U3 MI F aHHy11upyemc.11 npoU38eoeHueM 3/leMeHmoe, conpJVKeHHblX
C a 3/leMeHmaMu U3 r. Tozoa Oil.fl III060l0 HeHylle60lO uoea/la L KO/lbl(G s
KaJKOblu 311eMeHm U3 (ML n F)/FL aHHy11upyemc.11 npoU38eoeHueM 3/leMeHmoe.
conpJVKeHHblX c a 311eMeHmaMu 113 r.
aoKll3ame.m,emeo. TiyCTb u IF - KOHe'IHOnopO.>KJ].eHHbIA nOJ].MOJJ.YJib B Ml F.
Toraa cymecTsyeT aneMeHT x = a 81 ••• a8•, rae g,. E r, TaKOtt, 'ITO U x $ F.
Orc10aa-
(UL + F)(L + X) $ ULX + FL $ FL + FL,
H JieMMa cneayeT H3 Toro, 'ITO M JIBJI.lleTC.ll 06'heJJ.HHeHHeM nOJ].MOJJ.yJiett u.
JleMMa AOKa3aHa.
JleMMa 6. flycmb r - KOHe'iHonopo)Ki)eHHa.11 Mema6e11eea zpynna KOHe'iHOlO
cnel(UOllbHOW pama. Ec11u zpynna r He .11e11.11emc.11 no'imu HUllbnomeHmHoiJ, mo oHa
cooepJKum a6e11eey HOpMallbHYIO noozpynny 6e3 Kpy'ieHu.11 A maKy/0, limo Li r(A) =
= 1 u (j)aKmop-zpynna r / A - nOIIUl(UKIIU'ieCKO.R.
l10K0.3ame11bCm60 npoBeAeM HHAYKQHett no ro(r). EcJIH rpynna r He nO'ITH
ISSN 0041-6053 . YKp. Mam. xypH., 1993, m. 45 , N' 9
0 PA3PEIIHIMhIX rPYTITIAX, .. . 1279
HHJibnOTeHTHa.ll, TO Otta HMeeT HOpMaJibHYIO a6eneBy no.nrpynny 6e3 KpyqeffHjJ B
TaKyIO, 'ITO r / Cr(B )- Kotte'lttonopo)K,nettttag a6eneBa H D = !).r (B) * B. EcnH
D = I , TO yTsep)K.nem1e neMMhl O'ieBHAHO. Ilpe.nnono)KHM, 'ITO D * I. Tor.na, no
npe.nnoJIO)KettHIO HHAYKUlfH, cpaKTOp-r pynna BID HMeeT f'-.nonycTRMYIO a6enesy
no.nrpynny KI B TaKYIO, 'ITO !).r(K I B) = I. Ilpu 3TOM, O'ieBHAHO, no.nrpynny K
MO)KHO Bb16paTh rnK, 'ITO r .neAcrnyeT tta KI B paQHOHaJibttO uenpHBOAHMO. Tor.na
ueTpy .nno nony'lfffb cymeCTBOBfiliRe 3JieMe1rra g E C r(D) TaKoro, 'ITO CK 1 0 (g) =
l. O TCIOAa cne.nyeT, '{TO A,= [K , g] - HOpMaJibHM no.nrpynna rpynnbl r a A I n
D = 1. Ilo npe.nnoJJO)KeHRIO HHAYKURH q>aKTop-rpynna B I A 1 RMeeT r -
n;onycTHMJIO no.nrpynny A TaKYIO, 'ITO !).r(A I B) = 1 H cpaKTop-rpynna r / A
nOJIHQHKJJR'ieCKaR. Tor·.na, O'ieBHAHO, A H 6y.neT HCKOMOA no.nrpynnoA. JleMMa
AOKa3aHa.
IieCKOHe'iHbIA q>HHHTHO-annpOKCHMHpyeMbIA MOAYJib M Ha3hIBaeTC.ll MHHH
MaJibHO 6eCKOHe'iHblM, ecJIH mo6otl: ero COOCTBeHHbIR: cpaKTOp-MOAYJib KOHe'leH.
TeopeMa 2. llycmb r - KOHC'IHOnOpO)KOCHHaJI Mema6e11eea lpynna KOHe'IHOlO
cne LflW/lbHOlO paHW, M - (jJUHUlllHO- ann poKCUM UpyeMblU ~ f'-MOOYllb MUHU
MG/lb/10 6eCKOHC'IHOlO paHW. ToliJa M - MUHUMG/lbHO 6ecKOHe'IHblU ~pl-MO
iJyllb, iiJe p - npocmoe 'IUC110 .
a oKa3ameAbC11l60. OTMeTHM BHa'iaJJe, 'ITO aAAHTHBHa.ll rpynna MOAYJIH M
n u6o 6e3 Kpyqe1m.ll, JIH6o 3JieMeHTapnaR a6eneBa p-rpynna. lleAcTBHTeJibHO, ecnH
t(M) -:t= 0, To M co.nep)l(HT 6ecKotte'ittb1A no.nMon;ynh T TaKoA, 'ITO T -
3JJeMeHTapHaR a6eJieBa p -rpynna H r(M /T) < 00 • Ifa nocnen;ttero CO0THOllleHH.ll
cne.nyeT, 'ITO I Mp n Tl< 00 , OTKylla IM I Mp I= 00, H, CJie,noBaTeJihHO, Mp = 0.
IToKa)KeM HHLlYKQHeA no r 0(f'), 'ITO M - 3JJeMettTapHaR a6eneBa p -rpynna;
OTCIOL(a 6y.neT CJiel(OBaTb, 'ITO M - MHHHMaJibHO 6eCKOHe'iHbIA ~pl-MOAYJib.
Tipe.nnoJIO)KHM, 'ITO M - r py nna 6e3 Kpy'leHHR. EcnH rpynna f' IlO'iTH HHJibnO
TeHTHa.ll , TO no JieMMe 5. 1 pa60Tbl [8] M co.nep)l(HT CB06o.nny10 a6eneBy nol(rpynny
6ecKotte'lnoro pattra F TaKyIO, 'ITO I rt(M / F) I < 00 , H, cne.nosaTeJibHO, AJJH mo6oro
npocTOro p E rt (M / F) ttMeeM IM I Mp I = 00•
TaKHM o6pa30M, MO)KHO npe.nnonaraTb, 'ITO rpynna r He RBJIReTCR nO'ITH
HHJibnOTeHTHOA. TiyCTb A - non;rpynna H3 r. cymeCTBOBaHHe KOTOpoA cne.nyeT H3
JJeMMbl 6. T or.na BBHAY TeopeMbl I AOCTaTO'IHO noKa3aTb, 'ITO MOAYJib M HMeeT
~A-KpyqenHe. Ecm-13TO ue TaK, TO cornacno CJ1e,ncTBHIO 2.1 pa6oTbI [IO] cymecT
sy10-i cno6ol(nh18 ~A-no.nMo.nynh F ~M H 3JieMeHT O * a E ~A TaKne, •no Ka)K-
1:\hitt 3JieMeHT H3 M / F anuynnpyeTC.ll np0H3Be/J;eHHeM 3JieMeHT0B, conp.si)KeHHblX C
a 3JieMettTaM1-1 1-13 r. Ollen1-1ano, cymecTsyeT npocToe 'II-ICJJO p TaKoe, 'ITO a E
E p ~A , H, cne.nonaTeJThHO, .nn.si n1o6oro g E r &.8 = a K + p ~A - ttenynenoA 3Jie
MeHT KOJibQa ~ ,,A=- ~A Ip ~A. ITycTh ft =F I Fp 1-1 ft 1 = (Mp n F) I Fp. Tor.na,
TaK KaK IM I Mp I < 00 , TO I ft I ft 1 ,~~no JieMMe 6 Ka)KAhIA 3JieMeHT H3 Fi
aHnyJJHpyeTC.ll Ilpol-13Be,nettHeM 3JleMeHT0B, COnp.s!)KeHHhlX C O 3JieMeHTaMH H3 r.
fI03T0MY M0AYJJb ft I-IMeeT ~ PA-Kpyqe1me, 'ITO neB03MO)KH0, TaK KaK, O'leBHAHO,
F - CBo6oAHhlA ~,,A-MOL(ym, . TeopeMa A0Ka3aHa.
JleMMa 7. llycmb [' - zpy1111a , k - KOHC'IIIOC TIO/IC , M - MUHUMa/l b//0 6ec
KOIIC 'l/-lbtil k r -MOOYllb , [' I - HOJ),\tallbf/GJI 1100lpym,a KOHC'IHOZO UHOCKCa 8 [' •
ISSN 0041 -6053. YKp. Mam. )l(JP"·• 1993, m. 45 , N• 9
1280 A. B. TYUlEB
Toli)a cytt4ecm6yem k r cnooMOoy.11b M' ~ M maKou, 11mo I.MI M' I < 00 u M' =
= (Bi=IMi, rn:e M i - MUHUMaAbHO 6eCKOHe'IHbte k r 1-MOOy11u.
aoKa:JamMbCmBO. 113 (11] cnen:yeT, 'ITO M - HeTepoB k rcMOAYJJb, no:nOMY
M con:ep)KHT k r cnon:Mon:y J1b V, MaKcHMaribHblA OTHOCHTeJJbHO csoitcrna I MI V I <
< 00, TiyCTb T- npasrui: TPaHCBepCaJlb non:rpynnbl rl 8 rpynne r . Torn:a k r
non:MOAYJlb en, e T Vt)~ M uyneBOlt, H, c11en:osaTeJ1bHO,
M ~ $ (M/Vt).
teT
(3)
Qqesun:uo, '!:TO Mon:y11u MI Vt 11u6o MHHHMaribHO 6ecKOHetJ:Hble, 11u60 notJ:TH
npOCTble, npH'l:eM BBHAY <:pHHHTHOlt annpOKCHMHpyeMOCTH MOAYJIJI M " COOTHO-
weHHJI (3) noc11en:uult cnyqalt ueso3MO)KeH. TaKHM o6pa30M, M ~ ®i=IBi, rn:e B i -
MHHHMaribHO 6ecKOHetJ:Hbie ktcMOAYJJH, npH'l:eM MO)KHO C'l:HTaTb, '!:TO Bi () M =
= M,- "I:: 0. Torn:a MO)KHO nOJIO)KHTb M' = EB7= 1Mi. JleMMa n:oKa3aua.
TeopeMa 3. flycmb r - KOHe'IHonopoJKOeHHaR Mema6e11eoa zpynna KOHettHOlO
cne14ua.11bHOW paHza, A(D - ee FC - l{eHmp, k -1wHe11Hoe 110.11e, M - MUHU
MaAbHO 6ecKoHettHblu kt-Mooy11b maKOu, 11mo Cr<M). ToziJa:
1. Ec.11u IA(Ill = 00 , mo r - KOHettHoe pacumpeHue c6o6oiJHou a6e11eeou
zpynnbl KOHe'IHOW paHza.
2. E c II u I A (DI < oo, mo cytt4ecmoylbm HopMa11bHaR noozpynna KOHettHOW
uHOeKCa l I ~ 1 U kl 1-llOOMOOYllb M' ~ M maKue, 'Imo M' = EB7=1Mi, zoe Mi
- MUHUMQ/lbHO 6eCKOHe'IHble k r i-MoOy11u u r I /Cr<M,.) = Ai>-. Hi, zoe H i -
ceo6oiJHaR a6e11eea zpynna KOHe'l1t0w paHza u A,- - Hemepoe Htn11unmyc.
aoKa3tlmMbCmB0.1. C11en:yeT H3 TeOpeMbl 2 paOOTbl [12].
2. 0TMeTHM, '!:TO BBHAY JleMMbl 7 rpynny r MO)KHO JaMeHHTb mo6olt ee
uopMaJJbHOlt non:rpynnoli Kouequoro H,tin:eKca. Tio:noMY, TaK KaK IAITTI < 00 u r
yn:os11eTBOpJ1eT ycJIOBHIO max - r , M0)Kff0 CtJ:HTaTb, '!:TO 1 HMeeT HOpMaJlbHYIO
non:rpynny 6e3 KpyqeuuJ1 A TaKyIO, '!:TO r / A - cso6on:uaJ1 a6e11esa rpynna,
Ar(A) = 1, uA con:ep)KHT 1-nJJHHTyc B TaKOli, '!:TO ec11u A "l::B, TO Q)aKTop-rpynna
A /B 6e3 KpytJ:eHHJI. Torn:a ssun:y yTBep)Kn:euuJ1 2.4,IV pa6oTbl [3] n:ocTaTO'l:HO
nOKa3aTb, '!:TO A= B. noKaJaTeJlbCTBO nposen:eM HHAYKUHeli no r0(1), AJlJI qero
BBHAY TeOpeMbl 1 AOCTaTO'l:HO noKa3aTb, '!:TO M HMeeT kA-Kpyqeuue. Tipen:no
JlOXHM, '!:TO M 6e3 kA-KpyqeuHJI. Torn:a no c11en:crnu10 2.1 pa6oTbl [10] cy~ec-
TBYIOT cB06on:ublA ~A-non:Mon:y11b F ~M " a11eMeHT O "I:: a E ~A TaKue, '!:TO
Kaxn:bl.A a11eMeHT H3 MI F auuy11upyeTCJ1 npou3sen:euueM a11eMeHTOB, conpH:lKeu-
HblX c a aneMeHTaMH 03 r. TiyCTb a= L;l3,-t,-, rn:e 13,- E kB, ti - a11eMeHTbl npasolt
TPaHCsepcanu B s A " X - un:ea11 s kB, nopo)Kn:enHbllt a11eMeHTOM Il,.13i. no
JJeMMe 2 cy~eCTByeT MaKCHMaJlbHblA un:ea11 L KOJibUa kB, He con:ep)KllU{Hlt
Hn;eaJIOB, COnpJl)KeHHblX C X 3JieMeHTaMH H3 1. 0rcIOn:a CJien:yeT, '!:TO kAL He
con:ep)KHT 3JieMeHTOB, COnpJIJKeHHblX C a 3JieMeHTaMH H3 r . TaKHM o6pa30M, AJIJI
JI1o6oro g E r <i8 = aK + kAL - ueuyneBOlt a11eMeHT KOnhua S = kA/kAL. TiycTb
ft= F/FL H ft 1 =(ML() F)/FL. Torn:a, TaK KaK IM /MLI < 00, TO 1ft IF I I< 00 •
ISSN 0041-6053. YKp. Mam. JICYpH., 1993, m. 45, N• 9
0 PA3PEWHMblX rPYnnAX, . . . 1281
no neMMe 6 Ka)KJJ.blf.i 311eMe1n H3 ft I airnyJmpyeTC.H npoH3Be1ie1u1eM 3J1eMe1noB,
conp.H)KeHHbIX C U 3JieMel-lTaMH H3 r . TT03T0 MY M0AYJlb ft liMeeT S-Kpy<leHHe,
'IT0 HeB03M0)Kll0 , TaK KaK, 0 'leBHJJ.H0, ft - CB06oAHbif.i S-M0AYJlb. TeopeMa
A0KaJaHa.
TeopeMa 4. llycmb G - /\rme '111onop oJli.aewwJ1 c/Jwmm1w-a1111poKcuMupyeMaJ1
pa3peumJ.tafl ipynna M.UflllJ.ta /lbltO 6ecK01te'l1-1ozo pa112a, cmyne11b pa3pewuMocmu
1wmopoii Ne npeebtutaem 'IUC/la 3. M = Fitt (G) - ee uu11bllom e1m11-1b1il paau1ca11.
Tozoa M - :;1101e11map11aJ1 a 6e11eea p - zpy nna, M = Cc(M), u M
MU/-IUJ.ta/lb /10 6eCKO/le'lllblil ~ pf'-MOOYllb, lOe r = GI M.
l(oKa3ame.1i bC 11l60. TTyCTb M - MaKCHMaJibl!a.H a6enena H0pM3JlbH3.H IT0A
rpynna rpynnh1 G. T orAa M - ~f'-MOAYJJb MHHttMaJlbH0 6ecK0He'lnoro paura. H
1-13 TeopeMbI 2 CJreJJ.yeT, 'IT0 M - :meMenTap1-ra.H a6enesa p -rpynna H M - MH-
HRMaJibH0 6ecKotte•rnh1f.i ~ Pf'- MOAYJJb, r JJ.e r = G / M . TTpen.nono)KRM, 'IT0 M *
* Fitt(G). TorAa cym ecTsyeT uopMa111,na.H u1-1n1,noTeHTHa.H non.rpynna N :;;; G
TaKa.a, 'IT0 Z(N) :;;; M:;;; N 11 r (N /M) < oo. lfa n ocnen.Hero coonmweHH.a cneAyeT,
'IT0 IM ()NPI :;;; 00 • Torn.a H3 MIUlliMaJibl-IOf.i 6ecK0He'IH0CTH M0/WJI.H M Bb!TeKaeT
M () NP = 1. Orc10n.a r ( G / NP) = oo , II, cneJJ.osaren1,uo, NP = I. lfa nocneAHero pa
se1-1c Tsa cneAyeT, 'IT0 IN I Ml < oo H, 3H3'1HT, IIN, N] I< oo, a TaK KaK rpynna MHHH
MaJibH0 6eCK0 He'IH0r0 paHra. 0 '-leBHAH0, He C0Aep)KHT Co6cTBeHI-IbIX KOl-le'IHbIX H0p
Ma.TlbHblX no/lrpy nn. T O [N, N] = 1. TaKHM o6pa30M, N = M. H, c 11e/l0BarenbH0, M =
= Fitt( G). P aneHCTB0 M = Cc(M) HBJIReTc .H H3nec nrhIM CB0f.iCTB0M IIHJibn0TeHTr-10-
ro pMttKana pa3pewHMbIX rpynn. TeopeMa A0KaJa11a.
B 3aKJIIO'le1rne 0TMeTHM, ' JT0 C n0M0I.J..lbl0 TeopeMbJ 4 H3Y'le1rne p aCCMarpttBaeMblX
B Helt r·pynn B 31Ial.fIITeJil,llOf.i CTeneHH CB0AHTC.Sl K H3y l.feHHI0 MHHHMaJibH0 6eCK0-
Hel.fHbIX M0JJ.yJief.i HM K0IIel.fH0 n op0)KAeI-IIIblMH MeTa6e11eBblMH r p y nnaMH K0Hel.fuoro
cneI..(HaJibH0ro p a11ra, a B TeopeMe 3 C0/\ep)KHTCR HHQ)OpMaqH.H 06 3TliX M0AYJI.HX,
K0T0paR n03B0JJReT noJry l.fHTb AeTcU i bH0e 0 IIHCaHHe K0lle'IH0II0 p0)KAeHHhlX Q}HHHT
H0 - annpoKCHMHpyeMblX pa3peumMhIX rpynn MHHHMaJibH0 6eCK0Hel.fH0r0 paHra, CTy
neHb p a3pewHM0CTH K0T0pbIX He npeBblWaeT l.fHCJla 3, n on.o6no T0MY, KUK 3T0 6bIJI0
CAeJiaH0 n [3] AJI.H pa3pelllHMblX MHHHMaJihll0 He n0JIIlllllKJIHl.feCKHX rpynn.
I. '-lep1<UX06 C. Ii . rpynm,1 C :Ja)lal!lll,IMH CBOACTDa~m CHCTC MLI now·pynn. - M : 1-layKa, 1980. -
384 C.
2. Robinson D. J . S. , Zang Z. Groups whose proper q uotient s have finite derived subgroups//
J. Algebra. - 1988. - ll8. - P. 346- 368.
3. Robinson D. J. S., Wilson J. S. Soluble groups with many polycyclic quoti ents // Proc. London
Math. Soc. - 1984. - 48 , N°2. - P. 193 - 229.
4. Roseblade J. £. Groups rings of polycyclic groups // J. Pure and Appl. Algebra. - 1973. - 3, N°4. -
P. 307-328.
5. Brookes C. J. 8. Idea.ls in group rings o f soluble groups o f finit e rank // Math. Proc . Cambridge Phil.
Soc. - 1985. - 97. - P. 27 - 49.
6. flail P. Fini teness conditions for soluble groups// Proc. London Math. Soc. - 1954. - 4 , N" 16. - P .
4 19 -436.
7. '-lap,mB. C. 0 rpynnax aBTOMOpqlH3MOB HHJlb00TCHTHblX rpynn // YKp. MaT. )KypH. - 1954. - 6
N°3. - C. 295 - 304.
8 . flail P. On the fi niteness of certai n soluble groups// Proc. London Math. Soc. - 1959. - 9, N"36. -
P. 595 - 622.
9. 6ypoaKu Ii. KoMMyTanrnHa.R a.n1·e6pa. - M.: HayKa, 1971. - 7W c.
10. Brown K. A. The nullstellensatz for centain group rings// J. London Math. Soc. - 1982. - 2, N"26.
- P. 425 - 434.
1 I. Wilson J . S. Some properties of groups inherited by nonnal subgroups of finite index// Math. Z. -
1970. - 144, N2 1. -S. 19 - 21.
12. Tyw e0 A . B. M HHHMttJlbHO 6ec K011eq11ble MO/lYJJH 11an JJOKaJJbllO noJJHI..IHK JIHqecKHMH rpynnaMH
Ko11eq11oro pa11ra // JV CHM0. no Te0pHH K0JIC I..I, a.nre6p II MonyncA: Te3. coo6w . - lTbBOB, 1990. -
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n o11yqe1m 24.09.91
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| id | umjimathkievua-article-5930 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:24:15Z |
| publishDate | 1993 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/da/017fd0d7e2e54518a6041f402b24d8da.pdf |
| spelling | umjimathkievua-article-59302020-03-19T09:21:13Z On solvable groups, all proper factor groups of which have finite ranks О разрешимых группах, все собственные фактор-группы которых имеют конечный ранг Tushev, A. V. Тушев, А. В. Тушев, А. В. This paper deals with finitely generated finitely approximable solvable groups of infinite special rank, all proper normal subgroups of which determine the factor groups of finite special ranks. Вивчаються скінченнопороджені фінітно-апроксимовні розв’язні групи нескінченного спеціального рангу, всі власні нормальні підгрупи яких визначають фактор-групи скінченного спеціального рангу. Institute of Mathematics, NAS of Ukraine 1993-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/5930 Ukrains’kyi Matematychnyi Zhurnal; Vol. 45 No. 9 (1993); 1274–1281 Український математичний журнал; Том 45 № 9 (1993); 1274–1281 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/5930/8543 https://umj.imath.kiev.ua/index.php/umj/article/view/5930/8544 Copyright (c) 1993 Tushev A. V. |
| spellingShingle | Tushev, A. V. Тушев, А. В. Тушев, А. В. On solvable groups, all proper factor groups of which have finite ranks |
| title | On solvable groups, all proper factor groups of which have finite ranks |
| title_alt | О разрешимых группах, все собственные фактор-группы которых имеют конечный ранг |
| title_full | On solvable groups, all proper factor groups of which have finite ranks |
| title_fullStr | On solvable groups, all proper factor groups of which have finite ranks |
| title_full_unstemmed | On solvable groups, all proper factor groups of which have finite ranks |
| title_short | On solvable groups, all proper factor groups of which have finite ranks |
| title_sort | on solvable groups, all proper factor groups of which have finite ranks |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/5930 |
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