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AnreSpa H normta, 10, NO 5 (1971), 570-586.
YAK 519.45
rpynn c OAHHM onpEAE/iaioiuHM COOTHDIIIEHHEM,HE-COflEP>KAlIIME CBOBOflHblX nOflFPynn PAHFA 2
A.A.4EBOTAPb
B Hacxoamefi pa6oxe yxasbiBaexca Heo6xonHMoe H flocxaxoiaoe ycno-
BHB onfl BJioweHHa Hea6ejieBofi rpynnw, He conepjKamefl cBo6onHHx non -
rpynn panra 2, B rpynny c OAKHM onpenejiaiomuM cooTHoiueHHeM. STOT
pesyntTar BMecxe c peaynfararoM, aHOHCHposanHbiM B.B.HfaMMSHOM C6D ,
naex HeoSxOflHMtie H nocxaxoHHtie ycnoBHfl ana BnoweHHa rpynnw, yaoBJieT-
Bopflramefl HexpHBaaJifaHOMy xomaecTsy, B rpynny c onHHM onpenenaroinHM
cooxHouienHeM. RpH flOKasarejifacxBe Hcnonfaayexca onacaHHe nonrpynn CBO-
6oflHbix npoHSBeneHHfi c o6i,eflHHeHHeM, He conepwamHX cBo6onHbix nofl -
rpynn panra 2, xoropoe Mowex npeacxasjiaTfa H caMOcroaxejifaHbifi HHxepec.
B CBH3H c sxHM OTMexHM pa6oxy Hi 3 . nocBaffleHHyio HayieHHK) nonrpynn
CBo6onHoro npoHaseneHHa flsyx rpynn c c^temmeHHofi nonrpynnoft.
§ 1. nonrpynnti CBo6onHbix npon3BefleHHft c o6i,eniiHeHneM,
He coflep*ainHe cBoSoaabix nonrpynn panra 2
CBo6oflHoe npoH3BezieHne rpynn n H 3 c o6i,enHHeHHOft nonrpyn -H
nofl // 6yneM o6o3Haiaxb fl * & . MaaecxHO C 2 J , ixof — a *U6. if /y *^ rfMeeT eflHHCXBeHHoe npencr<|BneHHe B
^ = ^ /S752 . . .^/ (1)
rne n,fe.'~i , S^ npHHannejKHx cHcxeM© npencxaBBxeneft npaswx KJIBCCOB
CM6JKHOCTH rpynnbi rf HJIH 3 no nonrpynne // , i = /,'
croauiHe sjieMeHXbi 5- ( -^/j./ He npHHazuiewax K oaHofl H xoft we rpynne
fl ana B , j = /,2, • •• , 1 ~ / . 4ncjio n. ? O HaawBaexc
MeHxa U H o6o3HanaeTca -c(ti) . SjieMeHx U naabiBaexca
KH HenpHBOflHMtlM, 6CHH C (.U )= ft ^2 H S^ H 5 H6 npHHanJI6>KaT K OZI-
Floarpynnbi rpynn c OAHHM onpeAenaromHM cooTHOineHHeM 571
Hofi rpynne /^ HJIH £ . HMeeT MOCTO cneflyioiuaa
/1EMMA 1 (fl . H.MojiflaBaHCKHii H43 ). F l y c T b (1) e c T t,
H o p M a n b H a a <}) o p M a s j i e M e H T a ^ e ^ ? , c(&) ̂ <2 .
T o r A a n H 6 o a c o n p a > K e H c H e K O T O p w M
I I H K J I H H e C K H H e n p H B O f l H M b I M 3 J I 8 M e H T O M ,
J i H 6 o o f l H o a n a H H o n p e a c T a s H M B B H -
n e
^ = V~'qu (2)0 '
/?r n e c C o ) = / > H o p M a n b H a a ( f o P M a a e c T b
t t, ... £ (% ;? •/) ' ^ / ' - n p e n c T a B H T e n b ^ ̂ = /, 2,,. . .
... K) , K 1 H -g H e j i e x a T , a z ^ H ^ n e » ; a T' ' ff S
B o f l H o f i r p y n n e / r H n H ^ 3 .
TEOPEMA 1. n y c T b G- — /4+S . E C J I H n o n -
r p y n n a C r p y n n H <? H e c o a e p w H T s n e -
M 6 H T O B , C O n p H » 6 H H b I X U H K J 1 H 1 6 C K H H 6 -
/^n p H B o n H M b i M , T O f l j i a n o a r p y n n b i (^
c n p a s e n n H B o O H H O H S c J i e a y r a i n H x y T -
1) C c o n p a w e a a c n o n r p y n n o f t Cf
r p y n n vi n n n H ̂
2 ) ^ a s j i a e T c a o 6 i > e a H H e H H e M B O S
p a c T a i o i u e f i u e n o H K H n o a r p y n n , K a >K -
f l a a H 3 K O T O P W X c o n p a w e H a c H 6 K O -
T o p o f t n o n r p y n n o f t H 3 //
flOKASATE/IbCTBO. Tax KBK £7 He coflep>KHT sjieMeHTOB, conpa -
xeHHbix uHKJiHHecKH HenpHBoaHMbiM, TO no neMiwte 1 ana nroSoro W6. C
im&o C(\fi/)^ / , Jin6o W onHoaHaiHo npeacTaBHM B snae (2) .
/IEMMA 2 . F l y c T b B b i n o j i H a r o T c a y c n o a n a
T e o p e M b i 1. E c n n < 7 a : e . C > , £(q )<^ / H u=UglJ -"ff /n p e n c T a s n e H H e B H n a (2) , T o C(i]g l>~ )= O .
flOKA3ATEJlbCTBO. lfcV-q = v/eC . Ecnn £(y/} = O , TO-i / -/ v1 , —<
iJ CL^-Q = / t£ / / ,0-ljQ^ =-tifZV . TaK KaK 4? H <£,. ne
B oanoft rpynne /i HJIH y? , TO jingo ^(Q- iJo lT'') — / , a Torzia
&(VQ V~ ) == O , jm6o 6 ̂ • VQt V ) ~ neTHoe HHcno
MO>KHO, T3K K8K t^ltnV ) - HHCJIO HeHeTHOB. EcjIH
572 A.A.4e6oTapb
ana neKOToporo a &C V oi)-Q — O , oxxyna Q- i)Qr IS « Vo tf~ . TaK
nan CJIOBO t^Q.V'1 neieTHOft nnHHbi, TO cnosa & (VQ l>~/)=-0- Hycrt ,— / / •" f '
HaKOHeu, \ = il" Q, V - npeACTasneHHe BHAB (2). Torna i>'gV-g=i> gi/,' fff f 0 ff> ' fff '
OTKyna Q'VG U~ = VC~ Q i> V ~f . IIpHMeHaa K nocJieAHGMy paseHCTByI l r^paccy>KfleHHH, nonyiaeM t (itQjl)' ) = O.
J1EMMA 3 . F l y c T f a B b i n o j i H a r o r c a y c n o B H a
~ a V , U,= l)~ Q1 /f* ' * 2 ff
c t a a n e H H a a n a i / , H Z B B H n e (2) . B o n n
T e o p e M f a i 1 H u = u~ a V , U,= l)~ Q V - n p e ni 1 /f* ' * 2 ffZ 2
. . . ' • » / ( " ' / • *J ' ' ^
=Vftf > e c n K /7Z >n, . B n o c n e a n e M c n y n a e* y 7 ^ . / _ - . / » ^*
flOKASATE b̂CTBO. .̂"2 _ _QV - npeflCTasneHHe saaa (2). Torna t> a V.-V aiS
• "'= v v~'w- . EcnH " ", otKyna
nacTH nocneoHe'ro paaeHCTBa aanncaHo CJIOBO Hernofi anHHbi. CJIOBO,
croamee B npasofi "tacTH, MOWBT HMerfa HJIH Hyjiesyio, HJIH seneTHyio nna-
ny. IlojiyieHHoe npOTHBOpeiae noKasbiBaer paaeHCTBO c( Z^ tt~ Q, ty tfj )~Q-
. Tax KBK t(Zi>'oL^t, )=O
H Sf H O He npHHaane>KaT K oflHofl n roft we rpynne /I HJIH 2? ,TO npH- ^ » f
V. V1 K HopviajifaHOMy BHAy &, aonwno uennicoM norno -
x.e. z^= ij V^ , t>j l), , a rorna S^ = Z'xr7_/^+z,- ,
t = /, £, . . . , n. .
JleMMa aoKaaana.
IlycTb renepb, raK xe xaK H B C43 , £^,z^,... - COBOKynHOCTfa scex
CJIOB, HrparomHX ponb V B npeflcraBJieHHH ajieMeHTOB via O B BHAB (2 ) ,
. O6o3H-ainM nepea ^7- MHoacecTBO scex
M8HTOB H3 C , ArtHHa KOTOpblX H6 npGBOCXOAHT HHC^a 2t(tt)+ / .
. Torfla u~ t'y , u= V V-
)=•/ . Ha eAHHCTBeHHocTH 3anHcH (2) cneAyeT, HTO a H Gf npn-
K OAHoft H Toft we rpynne /f HJIH j3 , a rorAa npa noMomn
2 H 3 nerpyAHO noKaaaTb, HTO C- aBJiaeTca nOArpynnoH rpynnw
C H CjC-C'+f . ECHH U€.Cj , TO C(tt)<- 2-£(tf-^f) H H3 JI6MM 2 H
/ x^ /3 cjieAyer, HTO i^. i/<^." 6. // . TBKHM o6paaoM, z/c?. i/~ asnaeT -
ca noArpynnoft rpynnu /7 HJIH ^5 , a I?. C. V-*f asjiaerca noArpyn -
nofl rpynnbi // . ECJIH cpeAH sneMenTOB V- ectb sjieweHT HaH6o;ifauieft
AJIHHW, TO C conpajKena c noArpynnofi Ha /? HJIH 3 . B HPOTHBHOM
Ilonrpynnbi rpynn c OHHHM onpeaenaroinHM cooTHOiueHneM 573
ciiyiae C asnaexca o6i>enHHeHHeM Boapacraiomeii uenoiKH nonrpynn.ica*-
naa HS Koropbix conpajKBHa c noarpynnofi HS /•/ . TeopeMa 1 noxaaaHa.
Una BcaKOii rpynnH G H HSKOTOporo MHOxecTBa M ee anen/ieHTOB
nepea \M] .HJIH {//^}g 6yneM oSoaHanaTfa-noarpynny rpynnw G , nopow-
MHOaceCTBOM M
TEOPEMA 2 . r i y c T f a n o n r p y n n a C r p y n -
c o a e p K H T a n e M e H T , c o n p a -
H e c o n e p > K H T c B o 6 o a H o f t n o a r p y n n b i
p a n r a 2 . T o r a a C c o n p a w e H a c r p y n
n o f t 'Cy C (y~. , a J i a K O T o p o f t o n p a a e n n H B o•
o a n o H S c n e a y i o i n H x y T B e p > K a e H H f t :
r~< s> Hn bi CT= n -a- B
C/ KG n e n p H H a a n e ? K a T K o a H o i i H T o f io 7 <f 2
e r p y n n e / 7 H n n B , // «J C , Oi' €. // , q*6. ̂ r ,
4) Cf = \H, , U\ , r a e /-/f = Cf nH , " - U H K J I H -
e c K H H e n p H B o a H M H n K ) 6 o H s n e M e H T
V £• Cf n p e a c T a B H M B B H f l e zJ—u^/LiZ A-&/-/f,
m, n. - u e j i b i e , n p n i e M S C J I H L(.L>) ==• £(u) , T o£. c
V H e H s n a e T c a i i H K J i H i e c K H H e n p a
B O f l H M b l M , H 6 C J I H t-) = / ^ J < 2 7 ^ 2 ' « . tff - H O p M a j I f a
n a a ( j i o p M a a j i e M e H T a i ' , t o t H!^ (a Toraa H
r p y n n e / 7 H n H 8 . K p o v t e r o r o , a €. fY ^ V €. /-/ r
flOKASATE^bCTBO. Ifo ycnoBHro rpynna C co^epMHT no
Mepe onuH aneMeHT W , conpajKeHHbiii uHKJiHnecKH nenpHBOAHMOMy
Meirry i) . Oycrb i^ - sjieMeHT HaHMenfaiuefi HJiHHbi cpeaa raKHx ane -
M6HTOB i/ , U. npHHannewHT rpynne ^T , conpajKGHHofi rpynne C
, TO
HHneicc nonrpynnbi tff B rpynne /^ -f^, : //^ - HHneKC rpymm //,,
B . B
VJ-H1 , TO SJieMBHTM <2f 6 BZ H & CC^OO., nOpO>KAaiOT
574 A.A.He6oTapb
B Cr cBo6oflHyio rpynny panra 2.
1. FlycTfa^/7 ; M^\=\_3 '. /-J~\=- 2 • HoKajiteM, HTO B
yxBepaczieHHe 3) reopeMH. OycTj> O G. /? , q £ 3
>" ' • PaccMOTpHM rpynny {H,,$f,fc}o .$^4, , $
H THK kaK \.rff '• Hj\ 2 , TO Hf - aopManfaHaa nonrpynna B /?f , a no-
TOMyOf&/yf . AHanorJiHHO, &2 £- M, . RycTb, aanee,
3Ji<?M9HT z)G.Cf aanHcaH B
rue ct£ - npencTaBHTejii. Knacca • CMeKHocTH rpynnu fl .no nonrpynne
H , t>£ - cooTBeTCTByroraafl npeacTasHTenb rpynnu -B , i=/,2,,.,,sc ,
£,-=O,S; J—/.2 . ECHH zf(£j; ̂ / , TO BKnroHeHHe l> ' e. {Hf , Gf,3,}g
HMeer MOCTO. riycTfa £(V) > / HjAJia onpeaejieHHOCTH,£/=£ = / . Ec-
HH a} rux-r y. H , TO cno&a ZJfff H a VO q nopoacflaioT B Of CBO -
6oAHyio rpynny panra 2. TaKHM oSpaaoM, Ct.h,cli£.M . AHanorHino,/» '
# e //B npOTHBHOM cnyiae CfV?,) miesT Ht>pMaJU.HyK> aanact Buna (3), -me
£7 =• £j = / , , H MM MoxeM npHMeHHTb HHZiyKTHBHoe npeflnono>KeHHe. Ec-
JIH &r= O HJIH £.£=* O , TO, yMHOxaa i^ cnesa (HJIH cnpaaa) Ha Q',
(ann az ) , MM cuoaa nonynaeM aanacfa (3), rne £,•— f ^ j— 7,2 ,
K noBTOpaeM paccyKfleHHa. TaaHM o6pa3OM, C1=- (^f,Q^,Q2\/j .Tax
/^ : ^3"Cx3^ : H^ = 2 , TO #,**Cf.
2. rtoKasceM, HTO ecnn C^r : >"/,D =/ ,
FlycTfa, nna onpe«eneHHOcTH, HHKnniecKH
j , HM88T HOpMaJlfaHyiO (|>OpMy
(4)
H . Ecnit aAa e.
cnosa a) 7= G a u.a , a)j=C^M-) u>1^.u nopojKflaioT B ̂ cBo6onHyio rpyn-
ny paHra 2. ripennonowHM, HTO a/i, <3-, ̂ - M . ECJIH cla/tfare. // , TO
cjiosa td
nopowaaiOT cBo6oaHyio rpynny panra 2. Ecnic , Hanorieu, q AjG7 $L H ,
da/TjCij^. // , ctk.^,^. H , TO caoSoflHyio rpynny paara 2 nopoiKfla -
KJT cnoaa u}j=aucL , u). = ti ' audu . nojiyneHHbie npoTHBope-ana
r, HTO fljia nro6oro ate. tf , tf(f/)= / ,ac^€.J^7 , T.e.
, H ecnn cLe. Cf , £(cl) = / , TO cle.
IlOarpynnw rpynn c OAHHM onpenenaroamM coorHoinsHHeM 575
3. IIpennojiOJKHM, HTO mis HeKoroporo UHKJiHqecKH HenpHBOflHMOro ane-
Menxa V G-C HMeeT MecTO
O(.e./7 a.G.3 , v=/,2,.. . , ~£ . (B HPOTHBHOM cnyiae* ' - /-
c aneMSHTOM 1} ). TloK.ax.eM, HTO i) ' e. \/~l i^\. Kpo -
MS xoro, HJIH nonxoaamero fy.€.H, H HeKOTopwx uenHx /7T,/i z)=u./
Flo Bbi6opy tt, £ ^- k . PaccMOxpHM cnoaa
EcaiH npn npHBeaeHBH CJIOB u)f H uL K HopMantHOMy BHfly HH 2/ HH i/
ne nomoTHTca raK, HTO HopManbHaa (JiopMa cnosa 6^ KOHiaerca 3neM9H-
TOM H3 ^ HnH HopMaJibHaa <}iopMa cd HaiHHaeTca aneMenTOM H3 /J ,
TO (j)f H tt^ nopoJKnaioT cBoSonHyro rpynny panra 2, KTO
o6pasoM, npn £=f<
HJIH ~ f = A i OTKyna V =• LiA . FIonyieHa 6aaa
t, »• k . Toraa SCJIH i>= & V, , Vi u '= /c- £ H , TO vu~= t vU
U A- ^ - URKHR^eCKH HenpHBOflHM H 1i'(lS/l,}<LCl>) .Flo
™npeanono*eHHio, , ,
. ECJIH JKB V^VV, , i£~ i/e.// , TO
4. IlycTi. W=./^OC"//3/ . . . C r i a 4 < x / - HopMajibnaa c[iopMa sne -
ECHH VV U - UHKJ1HH6CKH HenpHBOflHMblii 3JI6MSHT, TO HO flOKaSaHHOMy B
W= ti^/iU . ECJIH
TO, icaK cnenyeT HS iacTH 2, vVi/ e. { Hf , 3}ff > a Toraa . f l , •
IlycTb WU He HBnaeTca uHKnaiecxH HenpHBOflHMbiM H £(_V/u)^-f . Tor-
na B cHny BwSopa sneMeHxa u {.(_ y/U ) < £\ Y/) , H HHAyKuneft no
cnosa y/ nerKO noKaaaxb, HTO V/ € | ̂ U, //f } /-,- . ECJIH B
cnosa W B HopManbHoft (J>opMe of=cXjj./ = / , TO BMecTO WU cne-
oyer paccMOTpeTb WU~f .
IloKaaaHO, T^KHM oSpaaoM, HT^ ecna rpynna < ,̂ He conepJKHT ane-
M6HTOB flJIHHW / , TO Cf = {^,^7}^ • H BbDIOnHaSTCa yTB6p>KfleHHe 4)
TeopeMM 2. ECJIH »ce cymecTByeT q 6. C. , £(o) = -f , TO C=(uiQ,/-/\a ' J ' c * f*g
STOT cjiyiafl paccMOTpHM noapo6Hee. He orpaHHHasaa o6mHocTH, MO>KHO
576 A.A.4e6oTapb
i>, HTO (4) - HopManbHaa 4>opMa sneMetrra IL H g € A' . EcnHm / "n
M6HT CL U- IIHKnHieCKH HenpHBOflHM, TO &LL — U. /Z U , OTKyflB £>
,.J, n-f „ . ., ,7 "• u- rt-ZU. , a Torfla LSJ=\ , Z/j.^ > T-e- BbinonHeno yTBepw
4). Ecna £(.Qti) — / (STO BOSMOWHO, Korna ^f(fz<7 = -^ )i TO HMeer Mec-
to 3). FlycTfa t CQU) •?• J H G^ He aanflexca unmiKHecKH
Torna CiLt HMeeT HOpManfaHyio
e. H i TBK K3K B npoTHBHOM cnyiae gjieMeHT (
t^CujHenpHBOflHM H HMeer onuHy, MeHbiuyro, HBM tCuj , ITO npo -
THBOpeHHT BblSopy it . AHaJIOrHIHO, CC^O,,A,U,C( <£ // , H B KOHU6f£ n £ / 7
KOHUOB
^ o6o3HaHHM npeacxaBHTenfa KJiacca CMewHocrn rpynnw tf (HJIH
no noarpynne H , K KOTOpowy npHHaflne«HT sjieMenr ^ . Tax
4^/r £ ̂ • T° ^f'f^f= i- HO 4^4e// - r-e-= (o. h*. ) = &„ . Ho Toraa zuia nooxoflamero 'ft- e. H ^1,AS=A&/c /c
flane'e paccMorpHM cnoaa O^ /4., <^ <^r • • • ^x- *= ̂ ^7 ̂ ' ' ' <2A- ' J ̂ / ̂ ' " "
-= (jy~ rt,~ ) = QK • Ho roraa ana noflxoflainero A £ // <%/£ =
58 /z iZ., . . Flponojmaa B STOM oyxe, ana nioSoro /z€./^. nonyiaeM paaeH-/» /
CTBO OU.it *= noQU , ftpG- Mf • MM noKasann, HTO B paccMaxpHBae -
MOM cnytae Cf*-\_l}—GM,Q, ^J-y. npnieM H} o C , T.e. HMeeT MecTO*!.y.TBep»cfleHHe S) . T.eopeMa AOKaaaua.
jj
CHEflCTBHE 1 . E c n H Cc-Cr—tf* 8 H a £ 7 B b i n o j i -
H a e r c f l H e x p H B H a J i b H o e T o * a e c T B O , T O
a n a r p y n n w £ 7 c n p a s e a n H B O o n n o H 3
y T B e g j K a e H H H l ) - 5 ) i - e o p e M 1 H 2 .
rpynna G nopojKnaeTca oSpasyiomHMH a , cf , ,
KOTOpwe cBasanS onpenenaroiniiMH cooTHouieHnaMH /? • (a-) , TO MH 6y-
fl6M TOBOpHTb, HTO G HM66T SBflaHHe G= < &7 , G^ , . . . ' /? • (O'.~)>.
ripHBeneM npHMepw rpynn, yflOBneTBopajraraHX ycnoBnaM 3) - 5) cnea-
CTBH3 1.
a) G = <aj; a-W*>.
Jlerxo BHaexb, HTO B G BbinonnaeTca HeTpuBHanbHoe
HanpHMep, x y JC y —/. G noniHHaeTca ycJioBHio 3) cnencTBHa 1.
rpynn c OHHHM onpeaejiaioiuHM cooTHoiueHHeM 577
6) C?'— ^G.o • d =~u > , C — \& , £Zu \ - cBo6oAHaa a6e -' ' *• * J (_f
jieaa rpynna panra 2. Ana nee BbinojiHaerca ycnoBHe 4)
B) tf = <a, h, ; <2^=/ , &/L = /LOZ, > ;
£ = /?"* 8 ;
~ r 2 P2 ZP2 2C = [a 6 a 6 a, , t
B C BbinoJiHaerca TOwaecrBp jc y jc iJ = / H ycnoBHe 5)
cneacTBHa.f/
adeneaoft noarpynnw £/C^=,
ycnoBHio 2), MO»;HO Haftrn B
OycTb /7 H 3 - HBB H3OMOp(J)Hbie noarpynnw rpynnw G H <£ -
H3OMOp({>H3M /? Ha -5 . Hycxfa Cy - rpynna, nopo>KAeHHaa o6pa3yro -
HIHMH rpynnti G H HOBbiM oSpasyiomHM o , KOTOpwe csaaaHbi onpene -
cooTHomeHHaMH G BMSCTG c cooTHOiueHHSMH zf^zzf = &<> HJIH
Bcex a. €. /7 . Ana &7
Q,~(G,t ; tat'*-
C/lEflCTBHE 2 . B c a K a a n o a r p y n n a ^ 7 r p y n -
n bi G , H © c o n e p > K a i i i a a c B o 6 o a H o f t n o n —7
r p y n n bi p a H r a 2, c o n p a j K e n a c H e K O -
x o p o f i n o n r p y n n o f t C a n a x o T O p o f l%. / "
c n p a B e a n H B O o n n o H S c J i e A y r o m n x y T -
1) £7, - 6 e c K O H e i H a a u H K J i H i e c K a s i
r p y n n a ;
2 ) £ ? 7 - n o f l r p y n n a r p y n n b i < ^ ;
3 ) C , H B J i a e T C f l o 6 i > e a H H e H H e M B O O -
p a c T a r o u i e f t u e n o H K H n o n r p y n n , K a > K f l a a
H 3 K o x o p b i x c o n p a x e H a c n o n r p y n
n o f t H O / T ' H n H B ;
4) H M e e r 3 a n a H H e
„ n K ) c5) ^ =.a,yz ,5)] , r a e
S78 A.A.HeSoxapb
6) Cf = {u,£)} , r A eSDc /7 H n H £>e^3 . U - a n e
M e H T 6 e c K O H 8 H H o r o n o p a A K a, H A n a
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KIT HecoKpaTHMyio aanHct
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ne coaepjKHXca CBo6onHoft nonrpynnw paara 2, xo flna C cnpaseflnHBO
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K1 KJIH K2 . npHMBHaa K (6) HJIH (8) xeopeMy Kypoiua o nonrpynnax
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2. HMeex Mecxo 2). B SXOM cjiy^ae BbinonHaexca nynxx 3) CJIBACX -
ana. floxaaaxenfacxBO coanaaaex c noKaaaxenfaCTBOM cooxBexcxByromero yx-
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Ooarpynnbi rpynn c OAHHM onpenenaromHM cooTHOinemieM 581
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rpynnw
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B caMOM aene,
*= g . Oojibayacb npeoSpaaoBaHHaMH THUS C3], nna
_ rpynnbi C. nonynaeM saaaHHe
(18)
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rionrpynnbi rpynn c OAHHM onpeaenaioimiM cooxHoiiieHHeM 585
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S K B H B a / i e H T H b i :
a ) n o a r p y n n a C r p y n n bi c ^ O A H H M on-
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Tax K&K B rpynne (12) [aj = ̂ o j , TO ona asnaeTCH nonrpynnofi
rpynnw
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586 A.A.HeSoxapt,
TaKHM o6pasoM, HMeex MSCXO
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s a r p y n n a C , H e c o a e p m a m a a c B o 6 o a -
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n a c t B r p y n n y c O H H H M o n p e f l e n a -
r o m H M c o o T H o u i e H H e M , H e o 6 x o a H M O H
a o c T a x O H H o , i T o 6 w o H a H M e n a o f l n o
H 3 3 a fl a H H ft (12) , (13).
SaMexHM B aaKJironeHHe, HTO ycnoBHa, HeoSxoaHMbie H nocTaTOHHwe
ana anoKeHHa aSenesoft rpynnbi B rpynny c OAHHM onpeaenaiomHM COOTHO-
, aHOHCHpoaaHbi B C6I] .
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, non pyKOBOflCTBOM Koroporo BbinonHena axa pa6oTa.*
/ i H x e p a r y p a
1. A.KARRASS and D.SOLITAR, The subgroups of a free productof two groups with an amalgamated subgroup, "Trans.Amer.Math.Soo?150 (1970),227-255.
2. A.r.KyPOUl/ Teopna rpynn, MocKBa, HayKa, H3A.3 (1967), 223-224, 247-248.
J>. V/.MAGSUS, A.KARRASS, D. SOLITAR, Combinatorial group theory,B.Y, 1966.
. 4. fl.H.MO/lflABAHCKHfi, O HeKOTOpwx noarpynnax rpynn c OAHHMonpenenaioniHM cooTHOmeHHOM, Cn6.MareM. K., V J U f Jsf96 (1967), 1370-1384.
5. fl.H.MO/IJlABAHCKHK, O noarpynnax rpynn c OHHHM onpenenaio-IUHM cooTHomeHHeH flHccepxauHS, HsaHOBO, 1968.
6. B.B.HEWMAK, Some results on onerelator groups, "Bull.Amer.Matn. Soc1.1,1968,74,113,568-571.
HocxynKno 4 HWHH 1971 r.
