Primary graded groups with complementable non-Frattini subgroups
We describe primary graded groups (in particular, locally graded, RN-groups) with complementable non-Frattini subgroups.
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| Date: | 1999 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | Russian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
1999
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/4731 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510896675094528 |
|---|---|
| author | Dovzhenko, S. A. Chernikov, N. S. Довженко, С. А. Черников, Н. С. Довженко, С. А. Черников, Н. С. |
| author_facet | Dovzhenko, S. A. Chernikov, N. S. Довженко, С. А. Черников, Н. С. Довженко, С. А. Черников, Н. С. |
| author_sort | Dovzhenko, S. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:12:54Z |
| description | We describe primary graded groups (in particular, locally graded, RN-groups) with complementable non-Frattini subgroups. |
| first_indexed | 2026-03-24T03:04:17Z |
| 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.
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IICClJpa'I'I~IlIHeBI~IMH rloltl'pyrlrlaMH//BoIlpOCld a./ll'e6pbl (r'OMeJlb). -- 1999. -- 1S. - C. 84-89 .
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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-
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rloalyqello 01.06.99
ISSN 0041.6053. ,Ytzp. ~tam. ~'vpu., 199~, m. 51, N e 10
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| id | umjimathkievua-article-4731 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:04:17Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/55/99ba48b4bdeaca50ff17ae8c7e1a8b55.pdf |
| spelling | umjimathkievua-article-47312020-03-18T21:12:54Z Primary graded groups with complementable non-Frattini subgroups Примарно ступенчатые группы с дополняемыми нефраттиниевыми подгруппами Dovzhenko, S. A. Chernikov, N. S. Довженко, С. А. Черников, Н. С. Довженко, С. А. Черников, Н. С. We describe primary graded groups (in particular, locally graded, RN-groups) with complementable non-Frattini subgroups. Наведено опис примарно ступінчастих груп (зокрема, локально ступінчастих, $RN$-груп) iз доповнюваними нефраттінієвими підгрупами. Institute of Mathematics, NAS of Ukraine 1999-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4731 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 10 (1999); 1324–1333 Український математичний журнал; Том 51 № 10 (1999); 1324–1333 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4731/6163 https://umj.imath.kiev.ua/index.php/umj/article/view/4731/6164 Copyright (c) 1999 Dovzhenko S. A.; Chernikov N. S. |
| spellingShingle | Dovzhenko, S. A. Chernikov, N. S. Довженко, С. А. Черников, Н. С. Довженко, С. А. Черников, Н. С. Primary graded groups with complementable non-Frattini subgroups |
| title | Primary graded groups with complementable non-Frattini subgroups |
| title_alt | Примарно ступенчатые группы с дополняемыми нефраттиниевыми подгруппами |
| title_full | Primary graded groups with complementable non-Frattini subgroups |
| title_fullStr | Primary graded groups with complementable non-Frattini subgroups |
| title_full_unstemmed | Primary graded groups with complementable non-Frattini subgroups |
| title_short | Primary graded groups with complementable non-Frattini subgroups |
| title_sort | primary graded groups with complementable non-frattini subgroups |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4731 |
| work_keys_str_mv | AT dovzhenkosa primarygradedgroupswithcomplementablenonfrattinisubgroups AT chernikovns primarygradedgroupswithcomplementablenonfrattinisubgroups AT dovženkosa primarygradedgroupswithcomplementablenonfrattinisubgroups AT černikovns primarygradedgroupswithcomplementablenonfrattinisubgroups AT dovženkosa primarygradedgroupswithcomplementablenonfrattinisubgroups AT černikovns primarygradedgroupswithcomplementablenonfrattinisubgroups AT dovzhenkosa primarnostupenčatyegruppysdopolnâemyminefrattinievymipodgruppami AT chernikovns primarnostupenčatyegruppysdopolnâemyminefrattinievymipodgruppami AT dovženkosa primarnostupenčatyegruppysdopolnâemyminefrattinievymipodgruppami AT černikovns primarnostupenčatyegruppysdopolnâemyminefrattinievymipodgruppami AT dovženkosa primarnostupenčatyegruppysdopolnâemyminefrattinievymipodgruppami AT černikovns primarnostupenčatyegruppysdopolnâemyminefrattinievymipodgruppami |