Примарно ступенчатые группы с дополняемыми нефраттиниевыми подгруппами
Наведено опис примарно ступінчастих груп (зокрема, локально ступінчастих, RN-груп) iз доповнюваними нефраттінієвими підгрупами. We describe primary graded groups (in particular, locally graded, RN-groups) with complementable non-Frattini subgroups....
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| Published in: | Український математичний журнал |
|---|---|
| Date: | 1999 |
| Main Authors: | , |
| Format: | Article |
| Language: | Russian |
| Published: |
Інститут математики НАН України
1999
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/157236 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Примарно ступенчатые группы с дополняемыми нефраттиниевыми подгруппами / С.А. Довженко, Н.С. Черников // Український математичний журнал. — 1999. — Т. 51, № 10. — С. 1324–1333. — Бібліогр.: 15 назв. — рос. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860094788424957952 |
|---|---|
| author | Довженко, С.А. Черников, Н.С. |
| author_facet | Довженко, С.А. Черников, Н.С. |
| citation_txt | Примарно ступенчатые группы с дополняемыми нефраттиниевыми подгруппами / С.А. Довженко, Н.С. Черников // Український математичний журнал. — 1999. — Т. 51, № 10. — С. 1324–1333. — Бібліогр.: 15 назв. — рос. |
| collection | DSpace DC |
| container_title | Український математичний журнал |
| description | Наведено опис примарно ступінчастих груп (зокрема, локально ступінчастих, RN-груп) iз доповнюваними нефраттінієвими підгрупами.
We describe primary graded groups (in particular, locally graded, RN-groups) with complementable non-Frattini subgroups.
|
| first_indexed | 2025-12-07T17:25:07Z |
| format | Article |
| fulltext |
Y~K 519.41147
C. A. ~OB~ICeHKO (BpatlCK. ne]~. yIH', Poce~la).
1~ C. tIepHHK011* (Htl-r Ma'reMaTrlKrl HAH YKparlllbl, Kl~el~)
I I P I / I M A P H O C T Y I I E H q A T b l E F P Y I I I I b I
C ~ O H O J I H I I E M b I M H H E |
I I O ] I F P Y H I I A M H
We describe primary graded groups (in particular, locally graded, RN-groups) with complementable
non-Frattinian subgroups.
Hal~ej~eno on•c np~Map~o c'ryniuqac'rrlx rpyn (aoKpeMa. JIoKam,no cryniuqaert~x, RN-rpyn) i3 l~o-
no~molmtmMr~ iteqbpa't-rinietmtart nil~vpynaMrt.
HanoMHrtM c.ne~tyIotuee onpeI~eJ~em~e.
Onpe/xe.nteHrie 1 [1]. Fpynna G nasbtoaemc,~ lTpU~tapno cmynenr ec.au o
npou3oo.abno~ ee nocgzpynne, nopo~cgenno~ cgo)'zta conpa.,w.eHnbt.~ttt npu~tapm, t,~tu 9ne-
;ttenma~tu..ato6a.a om.auqna,q om ecgunut4bt nocgzpynna tcone~moeo uncgetcca cocgep~rum
ucmunnyto noSzpynny Kone,moeo un3eKca.
K.nacc npI~MapHo cTyneHqaTr~x rpynri ~aect,Ma tuHpoK. TaK, OH BK.rl-lOqaeT B ce6a
i~.aacc ZOKa.rtbHO cTynettaaThIX rpynn rI BMeCTe C HHM rice zrirlenHrae rpynm, i , Bce
K~taccta rpyrm K y p o t t t a - q e p m m o B a [2], Kaaccta ~OKa.abaO pa3petunMtaX, pa/mKazl,-
HmX (B cMl,lcne B. H. I'Is .UOKanI, HO KOHeqHbIX, tlOHHHTHO armpoKcnMnpye-
Mb~X rpynn , 6nHapno KOHeHHI~IX, 6HHapHO pa3pettInMr~x rpynn . 13o~ee T01"O, K.nacc
IIpHMapHo CTyneHqaTblX r p y n n BKJ-IIOqaeT B ce6~l K~IaCC 6HHapHO CTyIIeHqaThlX
rpynn , T. e. rpynn , B KOT0p~x KaTx]~me /~Ba 3JieMeHTa rIopO~K/~a~OT 2IOKa.rlbHO CTy-
neHqaTy~o no.arpynny. (Onpe~tenen~e 6m~apno cTyrtellqaTofl r p y n n ~ rlpHHa/~:le:~KtIT
H. C. HepHrlKOBy (CM., aanpla.Mep, [3] ) . )
B pa6oTe [4] I'IO./'IHOCTblO 0rmcaHt,~ KOHeqar.Ie rpynm,~, B KOTOpblX ~ono~nzer , tb~
Bce nonrpyrIma, He aea<amHe a no~xrpymie ~DpaTrmm (TaKHe nonrpynm,~ MOmHO aa-
3aa'l'h Heti'bpaTTrlHrleBbllvlrl). B pa6o'rax [5, 6] FIOJIHOCTbIO Ollrl':aHbt OT.rIrlanble OT
rto~trpynn~a OpaTTnHU :lOKaJ'tbHO IIOqTH pa3petuH~a~,~e (r~ BMeCTe C ~TI.IM .rlOKaJlbHO
KoaeqHrae rt JIOKaJlbHO paapemHM~,ie) r p y n n u c/xono21aaeMblMH aeqbpaTTrmHemaMa
nozwpynnamL
OCHOBHb~MI, I peay.nbTaTaMr~ nacToJ~tuefl pa6OTbI J~a~arOTC,q TeopeMm 1 - 6 .
Ha~e aza nporiaa0zbaofi rpynn~a G �9 ( G ) a J ( G ) ~ COOTBeTCTBemto ee
nozwpynrta O p a T r m m a nepeceqenHe Bcex ee rlo~rpyrtrl KOHeqltOI'O rml~eKca; ecnri
H ~ no~trpynna rpynrmt G, TO H a = ['~ H g. Bce ocTa~,Htae o6oaHaqemia g~G
CTaH/~apTH~.
TeopeMa 1. Hycmb ~na ep)'nnbl G e~monn~emca cneSytoucee yc,aoeue: ec,~u
G ~ dp(G) ~ J ( G ) ;~ 1~ mo naa~ymca ~oneqnbte ,~tHomecmoa M ~_ G \ d P ( G ) u
K c J ( G ) \ 1 ~ne.~tenmo~ matcue, ~mo nu6o no~epynna ( K, M ) r.one~ma, .au6o un-
Oerc I (K , M ) : J ( ( K , M ) ) ] 6ectconeqen. TozOa u monb~co mozcga G ~ �9 (G) u
raacDa.~ noDzpynna H ~ ~ ( G ) zpynnbt G c2ono.au.~ezta e ne~, rozcga G ~ .auSo
om.auqua.a om etgunutcbt ~no,ane qbar.mopuzye~taa ~pynna, .au6o tcutr p -
zpynna nopacgra > p , nu6o zpynna nop,~cgra p 3 outga G = ( ( a ) x ( c ) ) X (b) ,
etge I a I = [ b I = I c] = p, c b = ca, a b = a (p ~ npous~oabnoe npocmoe ~tucno).
HalIOMHRM, qTO BriocHe qbaKToprtayemofl Ha3mBaeTc~l rpynna. B KOTOpOII/~orlo~-
aze~a Kam~aa ee no~trpynna [7]. TeopeMa H. B. tlepmmomoia [7, 8] ~aeT no,Hoe
XOnCTpy~T~mUoe onnca~ne Bno~me qba~Topazye~b~x rpynn.
�9 BhlIloJiIlella npH qaffl'HqllOJt no/IJI, ep,~KKr POCCH~CKOI'O tlJOll]ta tlJyll/taMell'l'aJlbllldX HCCJleI[OBRI1Hll
0~alrr N-" 99-01-Or
~) C. A, ROB~EHKO, H. C, qEPHHKOB, I999
1324 ISSN 0041-6053. Ytep. ,~tam. ~.'vpu., ! 999, m. 51. N ~" 10
HPHMAPHO CTYFIEHqATbIE FPYHHBI C ~OHOfIH~EMblMH ... 1325
TeopeMa 2 [1]. l lycmb G - - npu~tapno cmynen,cama.~ zpynna. TozOa u mo.abKo
mozaa G ~ O ( G ) u Ka~Da.~ noOepynna H ~ ~ ( G ) ~pynnbz G aono.an.ge~ta
nea, rozOa G - - zpynna oOHozo ua ouOoe, yra~nnbtx e meope.~te 1.
TeopeMa 3. Ilycrnb G - - 6 u n a p n o cmynen,~ama.~ (8 ~acmnocmu, .aora.abHo
cmynen,~ama.~) zpynna, TozOa u mo~bro mozaa G ~ �9 ( G) u raz~Oaa noOzpynna
H ~= ~ ( G ) zpynnut G Oono~n.gezta 8 nea, KozOa G - - z p y n n a oOnoeo u3 8u0o8,
yraaanm,u" 8 meopezte 1.
I/I3 TeOpeMbl 3 nertocpeixcTaenrlO BbITeKaeT, HanpnMep, csxe~ymmee n p e ~ o ~ e -
HHe.
C.aeOcmoue 1. I I ycmb G - - 5 u u a p u o pa3peuiu.~taa zpynna. Toz~a u mo,~t,Ko
moz3a G ~ dp(G) u Ka~ba.q no3zpynna H ~ ~ ( G ) epynm, t G bono.an.ge,~ta o
herr, Kozcga G - - ~pynna ocgnozo us ouOoe, yKasaum,~x o meope~te 1.
Teoper, ia 4. llycm~, G - - R N - z p y n n a . Toecga u morn,too moz3a G ~: �9 (G) u
Ka~Oa.~ noOepynna H ~: Cb(G) zpynm, t G Oononn.~e~ta o ne~, ~coeOa G - - z p y n -
na o3nozo tt3 atttgoo, yKa3an~tbtA" o meope:~te 1.
H3 Te0peMb~ 4 HenocpeacTBenrlo Br:aTeKaeT. i-mnpHMep, c.ne~y~o~ee npe,~.rto~e-
HHe.
C.aeOcmnue 2 [1]. l- lycmb G --pacguKam, naa (o cztbtc.ae E. 14. 1Laomruna)
zpynna. Toe~a u mo/tbKO mozOa G ~ ~ ( G ) u Ka~a.~ no3epynna H ~= ~ ( G )
zpynnbt G 3onomt~e~ta ~ ne~, Kozcga G - - zpynna ocgnozo uz ~ucgo~, ),ra3anm, tx
meopezte 1.
TeopeBdt4 3 H 4 - - qacTm, le c.nyqarl TeopeMb~ 2. au~e.nenHue BBn/Xy HX aa.,~riOCTH.
T e o p e ~ a 5 [ I ] . Hycmb G - - 2-zpynna. Toecga u mom,~o mozba G ~: ~ ( G ) u
Ka~iga.~ no~zpynna H if: ~ ( G ) ep),nnt,~ G cgono.an.~e~ta o t~er~, ~ozSa G - ,au6o
om.~uttna.~ om et~tt/tUt4bt 9neztenmapnaa a6e.aeaa zpynna, nuSo ttur~ut~ecraa zpynna
nop.~cgtca > 2, /tuSo gpynna cguacgpa nop.~c~Ka 8.
HrDKe no/1 .nnHeflnO~ rpynnota rtomIMaeTcz rpynna , HaOB, top~Ho npe,acTaamna.q
r, mTprmaMrt naa HeKoropblM rlo.rleM.
TeopeMa 6. IIycmb G - - z p y n n a . Toz3a u mon~,~r moz3a G ~ ~ ( G ) ; zpynna
G ,~unefma u t~a,'Kcga.~ noOzpynna H ~ ~ ( G ) epynm,t G cgono.~n~e~ta o uea, ~o-
z3a G . - - . a u 6 o zpynna ~mopoeo tt/tu mpem~,ezo ottOa u3 meope:,t~ 1, au6o G =
= A X B t t noOzpynna A ~ 1 pa3:~o.~uzta ~ np,~toe ~WousoeOemte uneapuanmm, lx o
G noOzpynn npoemt, tx nopmgKoe, ~ucno tr no Icaz,,-OO;~ty q e ~ (A) , 3a uc-
Kmo~emte~t, 6btmb ~to~em, o3noeo, KOHeqltO U ozpaHutteno netcomopo~ rot~cmanmo&
ne 3aouc.~u{ea om q, a nocgzpynna B tcone~ma u pa3Ao:Kuzta a np.~toe npou3aecge-
rtue nocgzpynn npocmbtx nop.~cgKo~. (B c.ay~tae G = A X B zpynna G ~no~ne qbar-
mopu3yezta.)
(B c.riyqae, Korea a Teoper, te 6 B = 1, qnC.nO n p z m a x no~rpyrm-~noeKrIre.rtefl
COOTBeTCTBylOI_ReFO ee paa.rlo.~eHHJl CqHTaeTc.q paBmaM ay.rno.)
~oKaaawe.rmc'rBaM xeopeM 1, 2, 5 H 6 npegnotu.rteM .neB, tMra 1 a 2 a npe,a.no~emIe.
. l le~s la 1. llycm~, G - - rone~na.~ uum, nomenmuaa gpynna, K - - ee nocgzpynna
mara,% qmo tcaa~cga/~ noOepynna H D K 3ono.an.~e~ta ~ G. TozOa na~Oemc,~ aSe-
.ae~a noOzpynna T ~_ G c 9~eztenmapnbt~tu cu.aoecKu~tu npu~tapnbt~tu nocgzpynna-
.~tu m a m a , t~mo [G : TI II gl ! , K ~ T = 1.
,Roxa.~meatmmao. MOTKHO CqllTaTb, q.TO G ~: 1. B TaKOB, I c.ayqae Z ( G ) :/: I .
I'lycTb M - - no,arpynna, nop0w,~aeHHaa BceMH a.ner, mnTaMr~ npOCT~X nopa~arOB Ha
ISSN 0041,6053. YKp. ~lam. ~.'vpu, 1999. m. 51, N'-' !0
1326 C.A. }2OB.~KEHKO. H. C. t..IEPHHKOB
Z(G) . OqeaH/II.-IO, M snO.nHe qbaKTOpHayeMa n e e Crl.;IOBCKHe npHMapHme no/ i rpyn-
n~ 3aIeMeHTapH~m a6e.nenbl. 1-lyc-n, A ~ / l o n o . n H e m m K K A M B M H D ~ / l o -
rio.nHeHnei~ KM a G, T = (AD) G. OqeBr~/IHO, G = K (AD) rl K N A D = 1,
Z ( G ) N D = 1. T o r a a I G:ADI= I K I . 3Ha,anT, I G : T I [IKI! (cM~, nanp~Mep,
[2], TeopeMa 12,2.2). TaK KaK A ~ T H A I'1D = !, TO BBH~y .neMMU C. H. qepHrl-
KoBa (CM., rianpHMep, [9], .neM~,m 1.8) T = A x ( Tf"l D ) . HOCKO.m, Ky Z ( G ) A ( T A
N D ) G ~ Z ( G ) N D = 1, TO (T["ID) c = 1. C.ae/IoBaTe.nbHo, aBH/Iy TeopeMr~x Pe-
MaKa T zB.rlaewc,q no/~nPZMtar~l nporlaBe/leHHeM rpynn, rtaol~lop(.~HbiX T / T fq D , H,
3Haan"rl HaOMOpqbHI, IX A. Tor,aa, HOCKO.m, Ky no/Irpynrla A a6e.neBa c :~.neMeHTap-
Hbllvll, l CHJIOBCKHMH npuMapHUMH no/IrpynnaMm TO ~,t T TaKa.,q ~ e . JIeMMa ,aoKaaaHa.
I I p e / l a o ~ e ~ m e . IIycm~, G ~ z p y n n a , G ~ r u e G c3ono.anne~ta npo-
u3oonbnan noc~pynna H ~ dp ( G). ToeOa cnpaoecgmtotn c.negylou~ue ymoep~c3enun.
1. @armop-epynna G /ep(G) mwnne dpanmopuzye;~ta, u O ( G ) coonacgaem c
nepeceqenue;~L ocex N <~ G, O,an Komopbtx G / N enonne dpaKmopu3Te;~ta.
2. ]Inn npousoo/u,no~ ~oKanbno Kone~no~ nocgepynnbt H ~ O( G) epynnbt G
nop~lanu3amop N G (H) nona/zono noneuen u N G (H) ~ J ( G) = 1 (o ,.tacmnocmu,
B N J ( G ) = 1).
3. ,lln.~npouzeo.abnOZO a.ae.~tenma g e G\r < ~ u (g) Iq J (G ) = 1.
4. �9 (.G) coonacgaem c nepece,enuezt nenomop,,tx nop,~taAbnbtX Oenumene~ no-
ne~mozo unOe~ca ep)'nm,t G ; o ,~acmnocmu, J ( G ) ~ dp ( G ).
5. Echu i G'/'~(G)I= P ~ P , m o G / J ( G ) ~ m m m m e c n a n p-epynna, u
3nn tcamcgozo g ~ G\~P(G) G = J(G) >~
6. Ec.nu J (G) r ~ ( G ) , mo G/J (G ~ ~ z p y n n a omopoeo u.au mpembeeo ouc3a
u3meope~b~ 1 ; o nepoozt cny,ae [ G/CP(G)[ = p r P, oo amopo~ G / ~ ( G )
9ne;~tenmapnan a6eneoa epynna nop,~3tca p-.
,Hor, asameat, cmao. !" }~eIICTBI, ITeJIBHO, G / �9 (G) ~no.nHe ~aKToprlayeMa ~i~-
aY .neuM~,~ 6 [6]. /Ia.nee, ec.nH G / N nno.nHe ~aK'ropnayeMa, TO ~ CH.ny .neMM~ 3 [6]
dP(G/~l) = 1. Ho cor.nacHo .ne~,t~,te 2 [6] N ~ P ( G ) / N ~_ d p ( G / N ) rt, 3HaUnT,
r ~_ N.
2. ]Ie~CTBHTeJI~HO, Ka:~z:~a,q no~Irpynna ~3 NG(H), co~cpa<amaa H. ~ono~Ha-
eMa s G ~, 3HaqaT, abbey ~ e ~ C. H. q c p m m o ~ a ~iononH~eMa ~ NG (H) . HO~TO-
lvly, OqeBHllItO, Ka:,K./Ia.,q no/I rpynna ~aKTop-rpynnta N G ( H ) / H /lono.nHaeMa B tle~.
CJm,aoBaTe.n~ao, B~H/Iy reope~,m~ H. B. t-IeprmKOBO~l o BnO.nHe t~aKTOpH3ye~,mlX
rpynnax [7, 8] N ~ ( H ) / H .noKa.nsno KOHe~ua. T o r a a BBany JieMr,~ O. tO. IIIr, mffra
NG(H ) .rloKadll:,HO KoHeqerl. BO3bMeM rtpOrl3BO.rlbHble :~..,-ICMeI.ITbl g �9 NG(H) rI a e
�9 Na(H) \dp(G ). Hyca"~ D ~ a o n 0 . n a e H H e K (g , a ) s G . T o r a a m~aeKc I G : D[
KOHe~errri, 3Ha,-IHT, J ( G ) ~ D. C.ne/xonaTe.n~ao, (g) ~ 1"7 J ( G ) = 1. I'IO~TOMy
sna/ly nporizso.rmaoc'ra g NG(H) ~ J(G) = 1.
3. /Ieflc'rnwre.m, no, [g I < '=' s~r~tty .neMMb~ 7 [6], a nOTOt,,ty S CH.ny y'mep:,KaenHz
2 rmeTOaU.lero npen.no:~ear!a (g) r'l J (G) = 1.
4. B o ~ e M npoa3noJlbItl~ll ~1 ~.neMelrr g �9 G \ r BBH/Iy y rnep .~ / I enaa 1 na-
c ' roamero npea.nomeHaz noa rpynna ( g ) r ( G ) / r a o n o n a a e n a ~ G / O ( G ) c
no~ot!Ir~m HeKoTopO~ rto/Irpynn~ D g / '~ (G) . B CH.ny yT~ep.a~term,a 3 H a e r o a m e r o
I$SN 0041-6053. Yrp. ,~tam. a,.'vpu.. 1999, m. 5 I, N ~- I 0
HPHMAPHO CTYI-IEHqATbIE FPYHHhl C ~OHO.rIH~IEMbIMH ... 1327
npe~no~eHHa nopa~oK ~.rleMeHTa g KOHeqeH. CJIe/~OBaTeJIbHO, HHReKC I G : D gl
KOHCqen. I'I03TOMy BBH/~y TCOpeMbl Hyarmape art~eKc S G no~rpynn (Dg)G XOHe-
qen. HcTpy/~no BH/~CTI,, qT0 ~ ( G ) coBna/IaeT c nepeceqeHneM n0~rpynn. (Dg) G,
B3Jna, lX no BCeM g e G \ ~ ( G ) .
5. IIycaa, I G / d P ( G ) ] = p. BO3BHeM Hp0H3BOJIbHbla HopMasmrlbllt aenwre~b N
KoHeqnoro nH/~eKcarpynnbl G rl rlpOH3BOSlbHhl~I ~JIeMeHT g r G \ ~ ( G ) . HyCTb D
~ o n o s m e a a e K (g) B G a L = N 1~ D o. BBH~ty yrBepac~eHna 3 HaCTOZtaero
npe~noa(eHaa HH/Iexc. I G : D [ KOHeaeH. IIoaToMy BcneacrBne Teope~m HyanKape
HHaeKC I G : L I xoae~eH. Torzla, OqeBH~HO, L conep:~aTC:~ B r~axcm~a:mHO~ no/~-
rpynne rpynnLi G . CneaoBaTenbHO, L c_ ~ (G) . HOHaTHO, ~TO ~ ( G / L ) =
= ~ ( G ) / L . TaKHM o6paaoH, I G / L : O ( G / L ) I = p a, aHaqHT, KOHe'~Haa rpyrma
G / L HMeeT e~HHCTBeHHy~O MaKCrlMa.rlr~Hylo no~arpynny. HO3TOMy G / L ~ ~ H K J ' I H -
qecKaa p-rpynna. Tor~anocKon~Ky g L ~ ~ ( G / L ), TO G / L = (g ) L / L. C~e/Io-
BaTen~nO, G = (g)L~ HOaTOMy, C y~eTOM TOrO, aTO L C D G H (g) rl D a = 1,
G = (g) .s D G H L = D G. B Ta~oM c~vyqae D G ~_N. 3HaaHT, BBH~y npon3noaI~aO-
CTH N J (G) = D G. CneaoBaTenr, HO, G = J(G) X (g). Ya-~epz~<~enHe ~oxaaaHo.
: 6. Hycaa, J(G)..~: ~p(G). PaccMowpma cny,cata, Kor~ta G / J ( G ) ~ ttHx~HqecKaa
p - r p y n n a . Tax xa~ J ( G ) ~_ ~ ( G ) (cM. yTBepacaeane 4) , TO, OqeBr~tnO,
~ ( G / J (G) ) = ~ ( G ) / J (G) . 1-loaToMy I G : ~ (G) ] = I G / J (G) : 0 (G) I J (G) I =
= I G / J ( G ) : ~ ( G / J ( G ) ) [ = p.
HyCTb G / J ( G ) He JtBJDteTCJt npnMapH0~l HHKJIHqeCK0~ r~pynnoll. Tax XaK
�9 (G) ~ J ( G ) , TO BBH/Iy yT~epa<neHHa 4 HaCTO~IIIICFO npe/Ino:~,cenrla Halt~eTC~I
nopManbHU~ /~enHTenb N KOHeqnoro HH~teKca rpynma G , z~na KOTOpOro
�9 (G) ~ N. 3aqbH~capyeM ero.
IIyCT~ M ~ rlponaaO;qbHbI~l nopMaJlbHht~/~CnrlTe~qh KoHeqHoro rlrlReKca rpynn~
G n L = N ~ M N ~ ( G ) . BBn~ty Teopema Flyauxapc aH~teKC I ~ ( G ) : L I KO~e-
qen. Bcne~tCTBHe Teopelvlbl H. B. ttepHrlKOBO~ 0 BrlOTIHe t~aKTOpH3yeMblX rpyrmax n
y T B e p ~ e n n a 1 Hacr0amero npe~noe~eHnJ~dpaKTop-rpynna G / O ( G ) nOKanbHO
KoHeqna. CneaoBaTenbHO, qbaxTop-rpynna G / L , 6y~yqH pactuHpenneM Koneqnoil
rpynnb~ c HOMOII.[blO a-lOKaJ~bHO KOtleqHOI:I, JIOKadIbHO KOHeqHa.
HOC~On~Ky, oqem~aHO, di)(G/L)" = O ( G ) / L ; TO ~ ( G I L ) ~ G / L , d p ( G / L )
1 n c yaeTOM y r ~ e p ~ e n a a 5 l G / L : ~ ( G / L ) I ~ ~. ~anee, aaaay n e ~ t 4 [6]
xaacaaa nonrpynna H ~= di)(G/L) r p y n n u G / L aono~naena a Heii. Cze-
aOBaTen~HO, Bnrtay T e o p e ~ 1 [6] H ne~cn~ 3 [6] G / L ~ rpynna TpeT~ero ~Haa Ha
TeopeMu 1. 3HaqHT, IO(G/L)I = p e ? H (G/L) / ~ ( G / L ) ~ a n e ~ e r r r a p H a a
a6enesa rpynna nopa~Ka p2.
TaK ~ax IO(G/L)I = p H N / L ~ (1)(G)/L = d g ( G / L ) , TO" ( N / L ) rl
N (i) (G / L) = 1. Cne~oaaTenbno, nOCKOSmKy G / L ~ Hea6eneBa rpyrma nopa~Ka
p3, TO N I L 1 H, 3aaqHT, N = L ~ M. HOaTOMy B~n~y npoHaaOSmnocTn M
N = J ( G ) . TaKx~M o6paaon, G I J ( G ) ~ rpynna Tpea~ero mtxta nz TeopeMu 1".
HpeanoaceHne ~oioaaao.
Ha npc~ano~KeHr~a HeHOCpeRCTBeHH0 BHTeKaeTTaK0e cnetXcT~He.
C.~eOcm~ue 3. 17ycm~, G ~ dpunumno annporcu~tupye~ta,~ ~pynna. Toecga u
monbro moeDa: G ~ q~(G) u Kaar noOepynna H ~ ~P(G) epyyma G 8ono,a-
n.ae~ta o ne~, r.oeOa G ~ zpynna oOnoeo u~ mpex ouOos, y ~ n t - t ~ t x o meope~e I .
ISSN 0041-6053. Yrp. ~tam. ~.'vpu., 1999. m. 51, N ~ 10
1328 C.A. ~OB)KEHKO, H. C. qEPHHKOB
B CB~3H CO CJIC]~CTBHeM 3 3aMCTHM, qTO BBHIIy TCOpeMIM H. B. LlepHHKOB01~I IIpO-
HaBOJ1bnaa snoJIHe qbaxTop~ayeMas rpynna qbmlwrHO annpoKcnMnpycMa.
C.ae~cmoue 4. ~ Hycmb zpynna G ~ dp ( G) c cgono.a~ze~tbt~tu nocgzpynna~tu
H ~ O ( G ) coOep,,rum aAe~tenm g ~ O ( G ) , romopbla c ra~rOt, z~l csou~ conpa-
~en~bal nopo,'rOaem ronetmyto nocgepynny. To~tga G dpunumno annporcu~tupye~ta
u .~an.~emca epynno~ ocgnoeo u3 mpex 6u~9o8, yrasanwotx o meope~te 1.
,Ll[otca.~me.abc.mso. BOabMeM ZaxOfl-nH6yIlb aJae~leHT a e J ( G ) . Bari/Iy yTscp-
acJZc~n#l 2 npc/I~o)zenu~ (g , ga) 1-) J (G) = 1. HOaTOMy, o~esn~no, (g, ga) = (g) ,
T. e. a e N c ( ( g ) ). CJzeIIoaaTem, uo, cnosa saH~ay yTBep)zzteu~a 2 a = 1. T a x e s
o6paaoM, J (G) = 1 a, aHaqnT, uacTosmee c~e~tcTaee cnpaac~auso BSr~y cJze~-
CTS~S 3.
Ha cJ~e~c'r~H,~ 4 sbrr~KaCT TaK0e CJ~CRCTB~4e.
C.ae~cmsue 5 [1]. l l ycmb G - -6unapno Kone~ma~ epynna. Tonga u mo.abKo
moeSa G v~ O ( G ) u ra.~Sa.~ noOzpynna H ~= ~ ( G ) zpynnbt G Oono.nn~e~ta s
net~, roecga G - - epynna oSnoeo uz mpex ouOoo, yrazannb~X o meope~te 1.
JIenMa 2. [lycmb G ~ 2-epvnna, A <3 G u A v~ 1. B ~ rone,~na.a nocgzpyn-
na epynnbt G u A [") B = 1. ToeOa C A(B) ~ 1.
,~o~aaame.4bcmeo. ,/~et4c'I'BI4TeJIt, HO. nycTt, C A (B) = 1. H ~ aai46oJ'l/:,Iiiaa
c p e ~ no~rpynn X r p y n n u B, ~.na ZOTOptax C A (X) ;e 1, ~ K ~ H ~ no~rpynna
rpyI~ill,I B, ]IJla KOTOpOl~ [ K : HI = 2 ; g a a :-- HHBOJIIOIAHH COOTBeTCTBeHHO t[3
no;~rpynn K~ H ~i H N A (H) / H dpaxTop-rpynnu N G (H) / H. Tor~a no~rpynna
(g, a) zoHcqna, tl I ~ (g, a') rl (HN~(H)/H) ~_ (g, a). CJ~c~oaaTcJzsno, noA-
rpynnaL/H = Z((g,a) ["I (HNA(H)/H) OTJmqna OT c~mmmu. Oqeav,~o, L _c
NG(K) ~ L~A ~ I. CaeAOBaTe~H0, NA(K) ~ I. HO NA(K) = C,~(K),
noczoJ~bzy K rl A = 1. FIpOTnBOpeq~e. J-IeM~a ]loma3arla.
]l[ora3ame.at, cmeo m e o p e . ~ 5. ]locmamoqnocmb oqesn/ma. ~OKa)KeM neoS-
xoOu~wcmb. HyCTb G ~ O ( G ) , ~ za~ /xas noztrpynna H ~ O ( G ) r p y n n u G
/~onomi~teMa n Heft. BO3bMCM a~eMenT g r G \ O ( G ) , B crony yTsep)K/Ie~mfl 2 n 3
npe~J~oa~c~n~ C~(<g)) ~ J(G) = 1. CJIeROBaTe.UBHO, Sm~zy ~C~U 2 J ((7) = I :
~iOaTOMy BBHZ/~y C~IeRCTBI4.~ 3 G - - rpynna o~uoro ~a Tpex B~ROa na Teope~b~ I . B
c~yqae, KOFRa G ~ rpynna ncpsoro BHRa, ona ~IBJ'IR~TC~ 3JI~MeHTapHoI~ a6cJlCBOfl B
OH.fly TeOpeMH H. B. qepHrlKOaOt~. Hpn p = 2 rpynna TpeTbero Bl~/la, KaK H3Be, CTI-IO,
eCTb rpynna arIaz~pa. Teope~a RoKa3ana.
]l[o~asament, cmao meopeMbl 1. ,l~ocmamo,~nocmb c yqeTOM TOFO, qTO ~JI$1
sno~ne qbaxTopnaye~ota r p y n n u G ~ ( G ) = 1 (cM. [6], ~ e ~ a 3), ycTanaBmmaeT-
ca Tax ~r Kax S aoKaaaTem, cTae TeopeMu [4]. Heo6xotgu,~tocmb sandy yTBepac.ae-
nH~ 1, 5, 6 r t p e ~ o ~ e m l z n~eeT ~ecTo B c.qyqae, Korea J (G) = 1. l'loKa~eM, qTO
J (G) = 1. IIycTb J(G) ~ 1.
1 ~ B a b y .yrBepac, aerma 2 n p e ~ o ~ e n r t z no~rpynna (K , M) Oec~oue~na a,
anaqar , rmt~ezc [ (K, M) : J ( ( K , M ) ) [ 6eczoneaen. BOabMe~ a n e ~ e a r g e ( K,
M ) \ O ( G ) . Ban/Iy yTaep:~J~euaa 3 npe~tno~enrm ero nop.q~OZ zoneqeH a (g>
["l J (G) = 1. IIycr~, h - - npa~apmatt aJ~eMerlr I~a ( g > \ t b ( G ) r t p - - iiricJlo, no
ZOTOpo~y On npr~tapen. 3aqbnxcapye~ h ~ p . HoJ~o:~rnn F~ = ( K , M ) ["l J (G) a
F = F 0 = F l ( h ) . Bsn~ty yraepac.aean.a4npc/I~to~enua F I ~ ~ ( G ) . Bc~eacTaae
y r e e p x a e u m a 1, 5, 6 n p e n n o x e n a s , r e o p e ~ u H. B. q e p n a z o a o l t qbaxrop-rpynna
G / J ( G ) noza.rm.o zoneqna. I l o w r o ~ / . . n e z c I (K, M) : F~ I zone , ten. Cae~oaa-
ISSN 0041-6053. YKp. ,~um. ~'vpn,, ! 999, m. 51, h ~ 10
HPHMAPHO CTYI'IEHqATblE FPYFII'Ibl C ~OI'IOIIHflEMHMH ... 1329
TCJIbHO, HH]~0KC [ F l : J ( F I ) ] 6CCKOHeUCH. I I y c ~ nptI HeKOTOpOM i > 1 y ~ e onpc-
Re,rleHa tlHBapnaHTttaJl B F no]wpynna F/_ l ~ F 1 KOHCHHOFO Hl-I~eKCa. Tor~a Fi_ l
co]~ep~rlT HCTHHHyIO no~arpynny L KOHeHHOFO HH/~eKca. BBHlly TeOpeMbl HyanKapc
rlrtaeKc I F : L F I KOHeqeH. HyCTb Tcnepb F i /L F - - MaKCrlManbHa~ cpC~H rlCTrlrlHlaX
nonrpynn rpynma Fi_ 1 / L F, MHBapHaHTHbIX B F /L F. Tor~aa Fi_ 4 / F i ~ KoHeqmaia
rnaBHblfl cl~aKTO p rpynn~ F. Hozwpynna F i KoHeqHonopo~L/~eHa B cHny TeOpeMI~
IIIpettepa (cM., nanpnMep, [2], TeopeMa 14.3.1). HTaK, a F onpc~eneH y6taBammnl~
HHBapHaHTHbIfl p~R
FO = F D FI D ... D Fi D Fi+I D ... D Fr = A Fi
i=!
KoHeqnonopo~n;eHnbxx no~rpynn Kone~noro nn~eKca, Ka~C~R ~aICTOp KOTOpOrO
rnaBH~R (H, 3HannT, 6y~yqH KoneqmaM, pa3naraeTcn B np~Moe npoI~3Be~em~e H30-
Mopqbmax Me~c~y C060fl HpOCT~X rpynn).
3a~eTnM, qTO tlpoH3aonI,na• nozwpynna Y~ F i rpynn~ F / F i z~ononHaeMa B JllO-
6Oa nonrpynne X / F i ~ Y / F t B cnyqae, Kor~la Y ~ O(G) (n qacTnOCTa, B cnynae,
Kor/la h ~ Y). }IeflCTBnTenbaO, B aTOM cnyqae Y ~ononHneMa B G c nOMOttlb~O
HeXOTOpOi~ no~rpynnu D. BBH;ay neMMta C. H. qepHnKona D N X ~IononazeT Y B
X. 1-loaToMy, oaeBrtano, (D (7 X) F i / F i ~ononHzeT Y~ F i n X / F i.
2 ~ HOKa~KeM, qTO npH Ka~ao~ f anz npon3BOnbnoro q ~ p crinOBCKaa q -
rm;trpynna Q rpynn~ F / F i ane~eHTapHaz a6eneaa.
FlycTs N = Ne/FI(Q) H R - { a l a a Z ( Q ) , aq= I } . OqeBrtjano, R aneMen-
TapHaz a6eneaa. TaK XaK Q ~ F ~ / F i, TO BBnay ~eMMta OpaTTrmrt F / F i =
= ( F l / F i )N. TaK Kax ( F~ F i) / (F~ / F i) ( ~-- F / F~ ) ~ UnKnnqecKaa p-rpynna,
TO ~lna HeKOTOpO~I tmxnnuecKoti p-nolarpynma L~ F i ~ . N F / F I - ( F l / Fi)
( L / F i ) . CneaonaTen~no, nocKon~Ky F ~ O(G) n F t ~ ~ ( G ) , TO L ~ O(G) .
IIOaTOMy noarpynna R ( L / F i ) ~aonom~zeMa a rpynne Q ( L I Fi) c noMom~m neKo-
TOpOtt nonrpynma D. Oaenazmo, D ~tononnzeT R B Q, a nOTO~y D ~ Q n D fq
N Z ( Q ) = I. BcHnyaTOrO D = 1 H, 3naUnT, Q = R .
3 a. I'IOK~DKeM, qTO/~J'IH HeKOTOpOFO m ~ ~ Ilprl Ka:4C3IOM i > m cnnoBcKaa p -
no~rpynna rpynma F m / F i ane~enTapHaz a6enena.
HyCTb P - - c a n o a c x a a p-no;arpynna rpynn~ F / F t, coaep~atuaa (h ) F i / F i, A
aJ'IeMeHTaptlbl~ a6enea HopMaJ'IbHbll~ ReJIHTenb MaKcHMaJIbItOFO n o p ~ K a rpynma
P H t i = [P : A 1. TaK Kale Kayxza3l no/Irpynna rpynn~i P, co~Iep~amaa ( h ) F i / F i,
~IononHae~a B P, TO BSmly neMM~a 1 t i < [ h [ [. BOSbMeM m r N, aria KOTOpOro
tm = max( t~ l k~ N) . HyCTS i > m a (p --rOMO~opqbH3M F / F i aa F / F , , . Tor;aa
PC " CHnOBCKaa p-nom'pynna rpynn~ F / F , , , A ~ <_ Pr A ~ " ~neMeHTapnaa
a6eneBa r~ [ P ~ : A ~ [ < t i <-- t m. CneaOBaTen~Ho, I P ~ : A ~ t = t m = ti = I P : A I.
HO~TOMy, oaeBnano, P N (Fro/Fi) ~ A . CneaoBaTenbno. crmOBCKaa p-noz~rpynna
P A (Fro/Fi) rpynma F m / F i aneMeHTapaaz a6encBa.
3aqbrlKcripyeM ~rtcno m.
4 ~ I'Ioxa~KeM, wro npn Kaa~aOM i > m rpynna F,n!F:, pa3pcmnMa.
HyCTS aTOne TaK. Tor;aa zmz neKoToporo i > m rpynna F i , 1 / F i nea6encna.
ISSN 0041-6053. Yh'p. ,,am. ~ayp,. 1999. m. 51. N ~ I0
1330 c.A. LIOB~KEHKO. H. C. qEPHHKOB
OqeBH//j-IO, OHa He Co/~epT, CHT OTJIHqHblX OT e/~HHrlLl/d pa3pelllHMblX HOpMad'lhHbI X/~e-
~aTenel t . IIycTt, P ~ cH.rIOBCKa~I p -no~ t rpynna r p y r m b t F / F i , co~aePmamaa
( h ) F i / F i, H D ~ ~onoaHeHne K P B. F / F i, U - - oI~I~H H3 MHO.FKIITeJ'IeI~'I p a 3 ~ o x e -
�9 ItH~I rpyrlllbl Fi_ l / F i B np~Moe nporlaBe•eHHe npOCTUX no~rpynn; S - - CHJIOBCKa~I
2 - n o ~ r p y n n a r p y u n e Fi_ t / F i 14 N ~ HopMa~UaaTop n o ~ r p y n n u S a r p y n n e
�9 H I F i = (Fi_ l / Fi) • ( (h) F i /Fi) . TaK KaK ( [ P I, ] D [) = I n U cy6nopMaZbHa B
F]Fi , TOamI/~y JICMMta 2.13 [9] U = ( P N U ) ( D N U). B CHZy ~IeMMbl ~aTTHHI4
H / F i = (Fi_ 1 /F i )N .
IIOKa~KeM, qTO p { [ U 1. B caMOM/xene, rlycTt, p [ [ U 1. Tor/xa, y~lHTlaaa~, tlTO
cHnoscKaa 2-no~trpynna r p y n n u U a6ene~a (CM. nn. 2 ~ H 3 ~ ~oKaaaTenbcTBa), H
HcrIosmaya reopeMy ~ m . Yo:rrepa [10], K~accnqbmmpymmym KOneqHble npocTl, le
r p y r m u c a6eneBo~t CHnOaCKO~ 2-nozwpynno~, TeOpeMy H. HTO [1 1 ], onHc~iaam-
myra ace C~al~Toprl3al2rlH c/~ByM~I rlO/~l-pyrlrlaMrlTMHO~KHTe~IatMH r p y n m a P S L 2 (qn)
IIpH IIpOH3BOJIbHbIX q a P H n ~ N, H pe3y~ISTaT~ [12], B COOTaeTCTmm C KOTOp/~I-
Mrl (Konemt~e npocTtae) rpynma Ja H "tuna PH He HMetOT HeTptmrla~,m,~x qbaKT0pH-
3atmia c ~ByMJ~ MHO~KHTenZMa, XOTa 6t~ O/XHH ~t3 KOTOp~,~X npl~MapH~X~,, y f e x a a e M c a
B TOM, w r o D ~ U coBna~aeT CHOpManrl3aTOpOM B rpynrle U ee CH~OBCKO~ 2-r!o~t-
rpyrmra H [ U : D N U 1 = P # 2. OTcmaa Bt~TeKaeT, wro I ( H / F~-) : N [ - - cTerleH~
p H ~a~a XOn~aOB0ia 2'-nozwp. y n n ~ T~ Fi rpynn~a N, cymecTBymmeit BBr~ay Teope-
M~t H. I I Iypa (CM., HanpriMep, [2], TeopeMa 20.2.6) , N = S ( T / F i ) H H / F i =
= ( F i _ I / F i ) ( T / F i ) . Tor~a .nocKom, Ky n ~ r ~l Fi_! ~ r T O T
.~ (G) . C~ae~toBaTen~,HO, T / F i /~OIIOJIItY/eMa B H / F i c noMommo HeKoTopo~
no~xrpyrmza W. rloc~le/~HJiJt, 6yztyqH 6HlipI.',MapHoll ( { 2, p }- ) r p y n n o a , ~aZ~y TeO-
�9 pema BepHcail~a pazpemHMa. Tax Ka~ H / F i = W N , S _~ N H, 0Ye~H~HO, S ~ W ,
TO B~Hay mMMra C. A. qyHHXaHa (CM., HanpHMep, [9], mMMa 1.36) (S H/6 ) ~ W rt,
anaarrr, n o z r p y n n a (S e t 6 ) ~_ Fi_l / Fi paapetunMa. YlpoTrlaopeqHe.
HTaK, p { [ F f l /Fi[ . Hcnonl, ay~l ~ICMMy OpaTI~HH, zlerKo y6CRHT~Ca 13 TOM, qTO
/Ins npor~zsOnSHoro npoc ro ro q [ I F i_ I /F i[ B F i 1 / F i Hall/levca CHdlOBCKaZ q -
no~rpynna Q, HopMaYm3yeMaz "( h ) F i / F i. OYemH~n0,/~onommHHe K Q ( (h } F i / F i)
B H / F i amnaeTca/xonoaaeHHeM r, Q , Fi_ 1 / F i. TaKHM o6pa3oM, R~a npor~3Bo~-
Horo q [ [ Fi_ i / Fi [ CHnO~CKaa q -no / l rpynna rpynn~a Fi_ ~ / F i ~ononH~eMa ~ Heft.
rIow Fi_ 1 / F i paapemHMa m cg~y TeopeM~ (I). X o ~ a (CM., HanpHMep, [13],
rn . VI , npe~noxeHHe 1.10). l'Ip0THm0pe~He.
HanoMHHM, ~rro no TeopeMe F: Llaccenxayaa npH npoH3mo~aoM n a 1~ cTyneHn
paapemHMocrr~ rpynn MaTpHU cTeneHe~ _< n Ha~ nonm~m orpaHH~eHm m COmOKynHo-
cana (cM., Hanpmaep, [14], reopeMa 3.7). MaKCnMyM ~TnX cTyneHefl O6OZHa~mM y e -
pea ~(n).
~IIJl IIpOH3BOJIbHOIt pa3pemHMOlt rpynnH X qcpe3 d ( X ) 0603HaqHM CTyrleHb ee
pazpemr~Moca~,.
5 ~ I Iycr t , X H Y c X - - rtport~i;o~,Hiae HOpMaoai, H~e ~temrre~m r p y m u a F , co-
~epmamaecA s F t, ~ A KOTOpmx X~ Y ~ KOHeqHaJt pa~pemnMaa p ' - rpynna c ~.ne-
MeHTapH~ra a6edxemaMH n p a M a p H ~ H no~rpynnaMH.
FIOKameM, q-to d ( X / Y ) < ~(1) + 1 Z-ux l = [hi. r l y c ~ H = ( X / Y ) •
ISSN 0041-6053. YKp. ~am. ~'vpn., ] 999, m. 5 I, N e 10
FIPHMAPHO CTYFIEHHATblE FPYFII'Ibl C ]2OFIO.flH,qEMblMH ... 1331
)', ( (h) Y/ Y), R ~ rto/.trpynna OHTTnHra rpynnbl X /Y. Bc.rleZ~cTmm TeopeMbl
MamKe R pa3..qaraeTc~ B rtpaMoe Hporl3Be,rj;eltrle HeKOTOpblx ~.neMeHTapmax a6e.ne-
BbXX MmmMa.nbmax HopMa.nbm,lX Re.nnTe.rlefi U rpynrua H. TaK KaK X / Y pa3pe-
mHMa, TO R = Cx/r(R) (cM., uanpm, mp, [13], r.n. III, npezt.no~emm 4.2) , n riOTOMy
R conrlaztaeT c nepeceqemteM tteHTpa.nH3aTOpOB CX/y(U), Ba.a'rFax rlo BCeM U. 3a-
~HKcnpyeM HpoHanO:mHbfl, i U. Hyc'rb q ~ qHCnO, nO KOTOpOMy npnMapeH U, L
--KaKa,q-Hl,16y/lh MaKCnMa.qbHaa cpe~I,l HCTHHHbIX no]~rpynn U, HHBapHaHTHtaX OT-
HOCHTeYlbtlO ( h ) Y / Y . . f l e r K o BrtaeTb, qTO t U : L I < ql. ~a.nee, no~rpynr t a
L ( ( h ) Y / Y ) aono~aHaeMa B H c noMombm HeKOTOp01,~ no~rpy lm~ D . OqeBHanO,
D aono.nvmeT L B X~ Y. I'lo~TOlVly, KaK .rlel'KO BH/2eTb, D f " ) U L~orlo.rlH~leT L a U ,
B crl.ny ~ero D N U ~_ X/Y . Tor,aaa.rlz rtpoH3t~o.nmmro g e H (D f"l U) g ~_ X /Y .
TaK KaK [ D ('1 U [ = [ U : L [ -< q l TO ( X / Y ) / CX/y((D f") U)#) ecvecTBenH~vt 06-
paaoM BK.na,ar~maeTcu r~ rpynny GLI(q) . C.neaonawe.nmm, nmmy OTMeqenHo~ weope-
r, lt,1 tlacceHxay3a d ( ( X / Y ) ) / C x / y ( ( D N U) g) -< ~(l) . TaK KaK U ~ mmnMam,-
HblaHOpMadlbHblii/~esI~ITeJlb rpyIInbl H, TO U = ( ( D A U)g[ g ~ H ) ri, aaaqrIr,
Cx/Y(U) = Ag~H CX/y ((DNU)'~)" rIo~rot~y d ( ( X / Y ) / C x / y ( U ) ) < ~(l) , B crt-
.rly qero d ( (X / Y) / R) < ~ (1). Csm~ol~a're.m, uo. d (X / Y) < ~ (I) + 1.
6 ~ IIOKaXe~,l, wro ~aKTop-rpynIm F,,,/Fcz pa3pemm, xa (cTynerm < 2 ~ ( / ) + 3).
TaK KaK rpylma Fm/F i , i >m, paapemmm (ClVt. n. 4 ~ /XoKasare.nbCTBa) H ee Crl.nOB-
cKaz p-noal 'pyr ina a6esmtm (cM. n. 3~ ro UBri/Xy "reoper, ua O. X o m t a - X n r M e n a
(cM., imnpm, tep, [13], rJL VI, ripe~.noxemm 6.6) F m re, feeT pmx F,, ~_ N l D_ N 2 ~_
F i xapaKvepHc'rH,qecKux Itoarpyrm c a6c.ne~b~ p-~13atcropo~ Nl / N 2 n p ' - ~ a l c -
vopaMl,l Fm/N I rl N 2 / F i. I-lpur, mpluae Hompyrmh~ noc.rte/mttx a6e.neBu (CM. n. 2~
~i noTor, iy d(Fm/N ~) < ~(1) + I, d ( N 2 / F i )< ~(l) + 1 (ct, t. n. 5"). C.rteao,aTe.nb-
HO, d(Fm/Fi) <_ 2~(/ ) + 3 n, mmmrr, aml,ay npOHaBO.nbUOCTH i F m/,co paapemu-
Ma (C d(F,,,/Fo~) < 2~( / ) + 3).
Hepeflzmrq K 3aK.nIOqHTeJIbHOMy ~'rariy/.tOKaaaTeaIbCTaa.
7 ~ YlycTh B ~ Hartlvmi~bmHi-~ q~eH npo~taaoRIIOVO pz/xa rpyrmb~ F m ~Fro,
ItMeIOIltHfl B Itei, i KOHeqltbI|~ HHI~CKC, C ~ CJ'IeLiylOl/-[lIl~ 3s iiillvI qJIeH ~TOFO pJ~l~a n
T~ C ~ rio,arpyrma, COCTOJmla~ ~13 BCeX a.rleMeHTOa KOlleqribIX rlop.,q~KOB rpyrima
B / C . TaK KaK F m/Fo~ KoHem~onopoxaeHa, TO aal~ay reopeMu O. IIIpei tepa
~mztrpynrta B H BMeCTe C xeM ([0aKTop-rpynim B / C KOaeqHorlopoX.L/~eIlhL T o r a a
riocro.nbKy B / C 6ecKoHe'-lHa tl a6e.neBa, TO B / C r T~ C. C.rleZ~oBaTe.nbHo, B~ T
OT.rLW.tHa..a OT c]lllllHIl, bl KOHeqHonopoh't/teHHa~ a6e.,qeBa rpynna 6e3 KpyqenH,q. Ho
B rpyrme ( B / T ) X ( ( h ) T / T ) / tonozHae~a K a ~ a a z noarpyriria, c o a e p ~ a m a a
( h ) T / T , H. aaam.tT, imrmy .rteMr, mt 8 [6] B / T = 1. rlo.ny,-mHaoe rlpoT~mope,-me
noKaar~maeT, ,.lTo J(G) = 1. TeopeMa aoKaaatia.
]Ioxa~ameat, cmao meopem~,t 2. HeoaxoOu;uocmb. I l y c n , G ~: �9 (G) a
Kaxaaa r~o/xrpyrma H ~ O ( G ) r p y n n u G ]Xorio.nHze~,m B ttefl. Boam, mta .mo6oli
a.neMegr g r G \ O ( G ) . BBr~y yTBepx,aeHHa 3 npe~,.aomennz g aMeeT gotte,talat~
nopa~lOK. HyCTh h ~ KaKO~-aa6y~Ib npaMapmafi aneMeax na (g)kdO(G).
Flosto~HM M = (h) . ,~astee, B npe,~nos~oxemta, qTO O ( G ) ~ J ( G ) *: 1, BOab~e~
ripo~iaBos~bmait aSmMeHT a ~ J ( G ) \ l . Ecm~ a e C a ( ( h ) ), TO no~toxnM K = { a} ,
a ec.nrt a ~ CG((h)) , TO a KaqecTBe K aOabMeM npOH3BO.rlbHOe KoneqHoe Mriome-
ISSN 0041-6053. Y~Ix mare. w.ypu., 1999, m. .5 I, N e I0
1332 C.A. I1OB)KEHKO, H. C qEPHHKOB
C~O o6paaymmnx no~rpynnm (h, ha> n J (G) . T o r a a B rl0pBOM c n y q a o ( K , M> =
= ( h > x ( a >, a BO BTopoM - - ( K, M ) = ( h , h a). OqearuXuo, 6eCmOHeqHOCTb no/I-
r p y n n u (K , M) BneqeT 6ecKoueqnocTh Hn~emca [ ( g , M) : J ( (K , M ) ) [ . CaeRoBa-
Te~bno, neo6xo~MocT~ cnpaae /vmaa B a ~ y TeopeMm 1.
~ocmamoqnocmb cnpaaemmBa Bandy TeopeMu 1.
~ o r a 3 a m e , a b c m a o m e o p e H t , t 6. H e o 6 x o c g u ~ t o c m b . BBrt~;y Teope~,t~
A . H , Ma.,"Ibl2eBa (cM., aarlpHMep, [ 14], TeopeMa 4.2) nporlaBo.m, Haz KOaeqnonopo-
~,,aeuHaz n o ~ r p y n n a rpynn~a G qbnnriT, O annpoKcHMapyeMa. IIoaTOMy scneacT-
Brte Teopema 1 G ~ r p y n n a ounoro na apex ee Bn~;OS. 1-IycTb G J~n.n.aeTca rpyn-
rlo~ rlepBoro Ba/la. Tor'/la BBrUIy TeOpeMu H, B. r"IepHtlKOaO~ OHa .rlOKa~bHO cBepx-
paapetm~Ma. C.ne~oBaTem, Ho. B cHny Teope~,fm 11.21 [14] G c o ~ e p m H T
rmBapriaH'rHylo noKa.mmo HHm, n0TenTHym n o ~ r p y n n y H, qbaKTop-rpynna G / H
no KOTOp011 KOHeqna ri a6eneBa. I lyc ' rb D ~ ~ononHeHHe K H B G . T o r ~ a G =
= H ),, D. B cayqae , Kor/ta H ~: 1, noaomnM A = H H B = D, a B c n y q a e H = 1
A = G rl B = 1. BBHay TeOpeMm H. B. qepnHKOS0~ n Kpr~Tepaz A. H . Manst~eBa
aaOMOpqbHo~ nPe/ICTaBriMoCTr! a6eneBot4 r p y n n ~ MaTprll2aMrI na/I no.rleM (cM.,
Hanpn~ep, [14], Teope~a2.2) A r~ B y~oB.neTBOpamTape6yeMUM yc.rtoBnJ~,L
~ocmamo,~nocmt,. Ec.nH G ~ r p y n n a BTOporo HnH apea'~,ero BH~a Ha Teope-
~ 1, TO B cmrr~ aTO~t TeopeM~ ~ ( G ) ~ 1 rt Ka,~z~aa nozxrpynna H ~ q~(G) rpyn-
n ~ G ~;ono.nH,qeMa B ae~i. I'lycTr, G = A >,, B. Tor~;a Baa~y TeopeM~ H. B. qepHH-
KOBO~a r p y n n a G ano.aHe qbax'ropuayeMa. 1-IoBToMy cor.rtacuo .neMMe 3 [6] �9 ( G ) =
= 1. ~a.nee,oBBr~y o r~e ' aennoro KpnTepHa A. H. Mam, ueBa A m m e ~ n a (B yKaaan-
HOM cMi,tcne), a nOTO~y Bc.ueaCTBHe KOHeq~OCTH nnaeKca I G : A [ rl G m m e a H a
(A. H. Ma.asUeB (CM., HanprlMep, [14], neMMa 2.3)). TeopeMa ~oKaaaHa.
Ha weopeMta 6 Bb~TeKaeT cae~yiomee npe~noa<eHHe.
CneDcmeue 6 [15]. Fpynna G ~,~,~emc~ mme~noft ano.ane dpatcmopuayez~oa
moeDa u mo.abt~o mozcga, t~oz~a ona npeOcma~uzta a ~uOe no,aynp,~toeo npou~t~eDenu,~
G = B X A ~one~tnog no3zpynn~ B, pa~o~uztof~ o np,~ztoe npous~eDenue nocg-
epynn npocmbtx nop.~tgtr u nocgzpynn~ A, pa3.ao~<u~oi2 a npa~toe tWous~eOenue
unaapuanmnb~x o G noDzpynn npocmb~x nop.~O~o~, ,~uc.~o ~omopb~x no ra~Oo~o'
p ~ ~t(A), sa ucrmoqenue.~t, 6~mb ~to~rem, oOnozo, roneqno u oepanuqeno ne~omo-
po~ ~oncmanmo~, ne aa~uc~u4e~ om p.
,ll~oKasame~at, emeo, HeoSxo~u~ocmb. I'lycTb G mmefl~a~ ~rtomm ~aKTopr~ay-
eMa~. M o ~ H o CqrlTaTb, q'ro G ~: 1. To r~a a crLny neMMU 3 [6] G ~: ~ ( G ) ri, alia-
tinT, Heo6xo/mMOC'r~ riMee'r Mec'ro am~ay Teopema 6.
Jlocmamoqnocmt,. ~e~,iCT~nTem, HO, r p y n n a G = B , ( A nnHet ina ri anonHe
qba~TopriayeMa amI~;y TeOpeMu 6 H TeOpeMu H. B. qepHnKOaOta [7, 8] o Bnom~e ~ a x -
roprtaye~b~X rpyrmax.
1. ,l~oa~ell~o C. A., ttepmur H. C. Fpynm,~ c ~tonom~:~eMs~Mn ne~pa'rrmmemaMn nom'pynnaMri I/
Me~w,apom,afl a~n'e6pa,qecgnfl ce~mlap, flocll/,.ILI/.e:lllllafl 70-~e'rum Kat~r ,btcmel,1 aJil'e-
6pra MI"Y: Tea. ItoKn. (MocKua. 10-12 dpeapaa~a 1999 r.). - M.: Ida/t..Bo MocK. yu-'ra. 1999. -
C. 23 .
2. Kapzanont~ M. H., Meps,~xoa IO. 14. Oc.oma "reopnn rpynn: - M.: HayKa. 1982. -- 288 c.
3. t'tepnuro, H. C.. Ma.aam, mta F. A. Oltuo yc.no~ne/tonoJm~e~oe't'n ~ rpynrtax 11 Y~p. Ma'r. ~ypn.
- 1996. -48, N ~ 10. -C . 1417-1425.
4. lloetuce,ro C. A. K reope~e H. B. qepunKoaoil o anomte qbaKropnzyer.,ax rpynnax // TaM ~ke. -
1999. - 51. N ~ 6, - C . 854-855.
5. ,l~oe.:,renro C. A..J'lolt~ll,no noq'rt.I paapetumnue rpynma c/torloJill~lr nr
nolu~ynna~s~//B'ropa~ Me~lwuapo/tua~ aan'e6panqec~a:~ Ko[l~peult l i . l l I~ YKpanue. noea~meu-
~ma naM~'m npotl~r J'l. A. Ka~yat,rma: "Fez. lto~L (Kue,: Bt4u,mla, 9-16 ~a~t 1999 r.). -
Bmmmta: l'lelt, y.-'r, 1999. - C . 73.
ISSN 0041-6053. YKp. ~vam. ~.'vp,.. 1999. m. 51. N e 10
FtPHMAPHO CTYI'IEHqATblE FPYIIrlbl C ~OFIO~H~IEMblMH ... 1333
6. Ztor~cenKo C. A. floKaJIbllO KOIICqllhle H JlOKa.~lhllO [1Otrl'H paapCHIHMIdC rpynnhz C ~onoJIIi~ICMIdMH
IICClJpa'I'I~IlIHeBI~IMH rloltl'pyrlrlaMH//BoIlpOCld a./ll'e6pbl (r'OMeJlb). -- 1999. -- 1S. - C. 84-89 .
7. Llepm~ooa (Eae~a) H. B. Bno~me qbaKTop~3yeMl~e rpynma 1/~[OKJI. AH CCCP. - 1953. - 92.
N -~ 5. - C. 877-880.
8. tlepuuxo~a H. B. Fpynma c ILono~maeMiaMx no/~rpynnaMa / /MaT. c6. - 1956. - 39, N" 3. -
C. 273-292.
9. tlepnuxoa H. C. Fpynma, pa30io~14Mi,te I~ rtpori3m~eRelirie neper rlo/irpynrt. - Krtet~:
Hayx. ]lyMKa, 1987. --206 c.
10. Walter J. H. The characterization of finite groups with Abelian Sylow 2 - subgroups / /Ann . Math. -
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11. It# N. On the factorizations of the linear fractional group L P ( 2 , p " ) / / A c t a Sci. Math. Szeged. -
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12. Monaxoa B. C. l-lpoxal3ejtelme cnepxpa3petunMofl rl n.HK./IHqCCKO~ HJIH npmaapuo~l r p y n n / / K o -
ue,4mae rpynma. - MruicK: HayKa H "rexHHKa, 1978. --C. 50-63.
13. Huppert B. Endliche Gruppen. I. - Berlin etc.: Springer, 1967. - 793 S.
14. Weh~'itz B. A. F. Infinite linear groups. - Berlin etc.: Springer, 1973. - 229 p.
15. ~el)ltuh'oa H. C., HuKumun B. B. 0 10~yx K~taecax JmHelam,tx neareJzemax rpynn//l'l'poraleMI.4 aJl-
t'e6pl,t rl KI46epIIe'I'HKH: Ma'repHaJu,I Me~ltynap. KOIt~. naMa'r~l a~a~t. C. A. Llyllleixlttla. q . 1. AJil'e-
6pa rl 'reoprta ,4rice~t: Te3. ItOK~L (FoMe~n,, 10-15 cenT. 1995 r.). - FoMe.nh: FOMent. yn-'l', 1995. -
C. 19-21.
rloalyqello 01.06.99
ISSN 0041.6053. ,Ytzp. ~tam. ~'vpu., 199~, m. 51, N e 10
|
| id | nasplib_isofts_kiev_ua-123456789-157236 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| language | Russian |
| last_indexed | 2025-12-07T17:25:07Z |
| publishDate | 1999 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Довженко, С.А. Черников, Н.С. 2019-06-19T21:52:41Z 2019-06-19T21:52:41Z 1999 Примарно ступенчатые группы с дополняемыми нефраттиниевыми подгруппами / С.А. Довженко, Н.С. Черников // Український математичний журнал. — 1999. — Т. 51, № 10. — С. 1324–1333. — Бібліогр.: 15 назв. — рос. https://nasplib.isofts.kiev.ua/handle/123456789/157236 519.41/47 Наведено опис примарно ступінчастих груп (зокрема, локально ступінчастих, RN-груп) iз доповнюваними нефраттінієвими підгрупами. We describe primary graded groups (in particular, locally graded, RN-groups) with complementable non-Frattini subgroups. ru Інститут математики НАН України Український математичний журнал Статті Примарно ступенчатые группы с дополняемыми нефраттиниевыми подгруппами Primary graded groups with complementable non-Frattini subgroups Article published earlier |
| spellingShingle | Примарно ступенчатые группы с дополняемыми нефраттиниевыми подгруппами Довженко, С.А. Черников, Н.С. Статті |
| title | Примарно ступенчатые группы с дополняемыми нефраттиниевыми подгруппами |
| title_alt | Primary graded groups with complementable non-Frattini subgroups |
| title_full | Примарно ступенчатые группы с дополняемыми нефраттиниевыми подгруппами |
| title_fullStr | Примарно ступенчатые группы с дополняемыми нефраттиниевыми подгруппами |
| title_full_unstemmed | Примарно ступенчатые группы с дополняемыми нефраттиниевыми подгруппами |
| title_short | Примарно ступенчатые группы с дополняемыми нефраттиниевыми подгруппами |
| title_sort | примарно ступенчатые группы с дополняемыми нефраттиниевыми подгруппами |
| topic | Статті |
| topic_facet | Статті |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/157236 |
| work_keys_str_mv | AT dovženkosa primarnostupenčatyegruppysdopolnâemyminefrattinievymipodgruppami AT černikovns primarnostupenčatyegruppysdopolnâemyminefrattinievymipodgruppami AT dovženkosa primarygradedgroupswithcomplementablenonfrattinisubgroups AT černikovns primarygradedgroupswithcomplementablenonfrattinisubgroups |