•qj hwweu oaxoiuaxbebxotr ox -bbs hwbh q8|ieh …math.nsc.ru/~alglog/01-10/08/08n4-6.pdf•qj...
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•QJ Hwweu oaxoiuaxBeBxotr ox
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xiiHHaHos ouoHh HU OHbanox :woeBdgo WHrnraXBairo Boxa
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B nocneasee speMa B nenaTH noaBnaioTca cooSmeHaa o TOM, HTO
onacan xoHenHbie npocTwe rpynnw c a6ejieBMMH CHJIOBCKHMH 2 —
nonrpynnaMH ( flOKaaarenbCTBO ne ony6mJxoBaHo). Hauia TeopeMa, xoKei -
HO, cneayer us Taxoro onncainia.
PaapeuiHMocTfa scex BcrpeiaioniHxcfl B SToft pa6oTe rpynn cneayer
us HasecTHofi p <%.* -xeopeMbi EepHcaftaa, nosTOMy anecfa He npHiunoct.
reopeMOfl ToMncona H ffreflra L6 3 , HO MeTonw pa6oxw
[ 6 ] oicaaanH Ha asropa pemaramee BnnflHHe.
McnonfasyroTca cneayroinHe o6osHaHeHHa: C., (X) H N,, (X)— ueHxpanH-
aarop H HopMannaaTOp MHoscecTBa X B G (naorfla anaHOK G- onycxa-#
erca). ECJIH X - rpynna, TO X oaHaiaeT X ̂ - i , kl (T) oaHanaeT MHO -
wecTBO noarpynn X TaKHx, HTO X/~) T =•! H J\f(X) ^ 7" ,l/1p (T) — MHO-
acecTBo Bcex p -noarpynn HS I/ (T) . •£ X > oaHanaeT noarpynny, nopo>K-
aeHHyio MHOXCBCTBOM X . SnaKonepeMeHHaa H CHMM6TpH«iecKaa rpynnw
creneHH n. o6o3Hana]OTca nepea /7 H S^ . HHBOJiiouneft HasbisaeTca
nopaflxa 2. Hepes O(H) o6o3HaiaeTca nponsBeaeHHe scex nop-
noarpynn HeieTnoro nopaflxa rpynnH H , qepes S(H) - HBH-
paapeniHMaa HopMa/ibHaa noflrpynna rpynnw H . I H I oaHaiaeT
nopanoK H .
B HacToameft pa6oTe Syaer noKasaHa cneoyromaa
TEOPEMA. K o n e i H a a n p o c T a a H e a 6 e n e -
B a r p y n n a n o K a a a x e J i a 3 0 K 3 O M O p c } ) H a
Ha nporajKeHHH Bcefi pa6oTbi G SyneT o6o3HanaTfa
TeopeUe npHMep.
2.
B STOM nyHKTe MM aoxaweM HecxonbKo npocTwx peaynbTaTos o crpo-
KoaeiHbix rpynn noxasaTena 30, y KOTOPHX nepaspeiuHMwe
jaxTOpbi H3OMOp4>Hbi /7- . B KOHue nyHKTa, Hcnoni.3ya
Bpayspa (C23 , 'npefljio>KeHHe 6 ) , MM noicaxceM, HTO nopaaox
2-noflrpynnbi rpynnw G He MeHbiue 16.
B STOM nyHKTe G osHanaeT MHHHMaJTbHbifl npoTHBopeiamnft reopeMe
npHMep. G - KOHeHHaa npocTaa Hea6enesa rpynna noKasarena 30, HeHso-
MOp4>Haa ft g , H nepaapeuiKMbie KOMnosHuaoHHbie (fiaKTOpbi jiioSoft co6cT -
B6HHO2 nOflrpynnbl H3 Cr E3OMOp4>HbI fl r .
462 B.fl.MaaypOB
JIEMMA 1 . E c n . H L / - K O H e i H a a H e p a s p e -
ui H M a a r p y n n a, n ro 6 o ft H e p a a p e u i H M w f i
K O M n O S H n H O H H b l f t <J> a K X O p K O T O p O f i H 3 O -
M O p $ S H ft , T O
U/0(U] ~Lf
flOKASATE/IbCTBO nposeflSM HHflyKuneft no nopaflKy U . MOJKHO
, «ITO 0(U) — i • ECJIH S(U)^=- / i TO no npeflnonoxeHHio HH-
0(S(U)) nextHT B 0(U] , noaxoMy 0(S(L7)) = i • TaK KBK CHnoscicaa
2-noarpynna T rpynnw i[7 KOMMyxaxHBHa, TO T^S(L7) < S(U) H
T n S(U) cosnaaaex B S(UJ co CBOHM uenxpanHsaxoptoM.
Ky Z/ x . . . x l_, nopoMcnaexcH CBOHMH CHJIOBCKHMH 2-noarpynnaMn
Tax KBK T sneMeHxapna, C" (TnS(U))=Lf* ...x t ̂ x (T n S (U)} ,
roe L{^/3g. (i=*i,...,k^. TnS(U)<S(U) , TO TnS(U)4U .,
C(mS(U))<>U * L,x...xLK4U . 0,eBHAHo, U=(Lfx...x£/<)S(C/) ,H xaK K3K (L,x... x L,^}^ S(L7)=-1, neMMa B SXOM cnyiae AoxasaHa.
7 t
FlycTfa S (U} — 1 . OSoananHM nepes M noKoni. rpynim U , TO
ecTfa npOH3BeaeHHe scex MHHHManfaHtix HOpManbHbix nonrpynn rpynnnw U .
M B HauieM cnynae - npa\ioe npon3BeaeHne Hea6eneBHX npocTux rpynn,
H , B nacxHOCxH, C (M) = 1 . Ecna M = U, xo ace noKasano. Flycxfa
M =^ U H nycxfa N - nonHbift npoo6pa3 B U HeKoxopofi noarpynnti
npocxoro nopaoKa HS /M . EcnvtN^U, xo no npennonoKeHHio
IIHH N=M X < JC->, rae^rc> - uHKnunecKaa rpynna npocxoro nopazuca H
C (MJ^ j[ . FIosTOMy JV= U H /vVf — uHKnHHecKaa rpynna npocxoro no —
/? . ITycTt £ - MUKHMantHan nopManrfaHaa B U nonrpynna. Ecna
M • TO £y/C
- rpynna nopanjca p . FIonHbifl npoo6pa3 -C^OB U HHBapaaHxeH B
U H conepjKHT no npeanonojKeHHio HHnyicann HopManfaHyro noarpynny.nso-
MOpi})Hyio<a:>I HTO npoTHBopeMHT ycnoBHio Sf£7) = ̂ . EfoaxoMyA/ -MHHK -
HopManbHaa noarpynna npocxoro HHfleKca p B U , a £7 HSO -
Oc.na6neHHaH npo6neMa Bepacattaa ana noKaaarena 30 463
MOp$Ka noArpynne rpynnw aBTOMOp$H3MOB /W . FlycTb & - aneue-HT jro-
p B £7\ , nycTfa Mf — MHHHManbHaa Hopwanfaaaa B M noflrpyn-
rra. IJo ycnosHio M — flg . ECJIH Mr=M , TO £7 — S,. , H
£7 paB8H 60. IIoaTOMy M^M . ECJIH Mf =M, TO M '< U , TSK xax
oaroMy M ,& =^= M . H < A/,S A/ >=A/ XA/,S
s'-'-^ <;c'+' '^1 ..... M* > , a A/, '^=<M ..... Mf>=M
. , . . , j v y , j c - .17 »'_ / ' * » . • 7 I ' 2 •'
, AT X... V MS ] =i , HAff"- .V/.Sx . . . \M.5'l *fjf H,' ' ' ' et+/ S1-"1"'
nepecraHOBOieH c nioSbiM ansMSHTOM HS A/. • B ^/
cymecTBy«r aneweHT y , HenepecTaHOBoiHtia c x., H
Ho [je,yUe M, , a C^r,yl] e A7^ ^
/Vf =M • 3TO osHa^aer, TTO 5 C HopMannayeT M^ , TO ecra. Sl c
H i—p — i> Ecnn renepb y - sneMenr napaAKa Z7 as M1 ,TO
, S> aaoMOp^Ha cnneTeHKK><c> \ s> , HTO
cnneT6HHa<y> X<S>paaeH p . Hrait, M = U , H
JIEMMA 2 . n y c r t C 7 - K O H e H i r a a r p y n n a
n o K a s a r e n a 30 H n y c T fa n to 6 o ft H e p a 3 -
p e IU U M bl ft K O M n O S H H H O H H H f t 4> » K T O p £7
H S O M O p t J i e H /? _ . r i y c T b 7 " — c H n o B C K a a 2 - n o n -
r p y n n a r p y n n b i t / . r i o a r p y n n a X H S £ / T O r -
a a H T o ji t. K o T o r n a J I B M C H T B O(.U), K o r-
f l a CX , T 3 H M e e r H e H f i T H b i f i n o p a a o K .
flOKASATE/lbCTBO. Ho
(U) ^ L, x . . . *LK x S iU)/0 (U) ,
rne Z/^^/7 (£=/,...,£) . EcnHX^-^^IJ) , TO nopaflojcCX,7"3
T6H.
ITycTfatlV^ InweeT ne^eTHNfl nopanoK. Ecnn k = O , TO B
CHJioBcxaa 2-noarpynna nHBapaaHTHa H coanaaaeT co CBOHM
POM. 3ro osHaiaeT, HTO
Tocu)/ "l ore/)/fleer) , /ocuu= focu) H cx,
IlycTfa k ^ j. H M£ -no^Huft npoo6paa B £7 noflrpynnw // X ... *• L^*
x^t+^ x ... X ^A. X S(UyO(U) . Toraa U/M; *./Jf H, nocxonwty B/7^
Her HeenHHHiHoft nonrpynnu He^feTHoro nopaaxa, ssaHMHtitt KOMMyraHT KO-
464 B.fl.MasypOB
TOpoii c CHjiOBCKoft 2-noflrpy nnofi us /J g. HMeer HeHSTHbift nopaaox,
X -4: A/ n, anaHHT, X 4: S ft77 . ECJIH 7^ -cnnoBCKaH 2— nonrpynna
as S (U) , TO [IX , 7~f II HMeex HeneTHbiii nopaaoK, n no ywe aoKaaaHHo-
My X4 O(S(LT)) ~ 0(U) •
/1EMMA 3. n y c T b £^ - K O H e i H a a r p y n n a
n o K a a a r e n a 30 H Z> - T a K a H H o p M a n t -
H a a n o n r p y n n a a C / . H T O ^/2/ — /7^ H n H /? -.
T o r f l a U = LF , r n e P^ % .
flOKA3ATE/IbCTBO. HnayKUHH no nopaflKy rpynnw. ECJIH L ae
HHnfanoTeHTHa, TO nycTt, P — HenHBapnaHTHaH cHnoBCKan p — noarpynna
rpynnw L, ana HeKOToporo /? . LT=LN(P) H \J*f(P)\4. \U\ Kpo—
Me Toro, - , ornyna - , ( rfle
/• — /A . MM nonyiaeM B STOM cnyjae HCKOMoe,
U =
ECJIH Zv HHJlfanOTGHTHa H L = \_Ll , L J ̂ ̂ , TO /^// = /L ' ' /L ' '
1 / L — /L . FIojiHbift npooSpaa F noarpynnw // L B U HMGBT
kiHii nopafloK, H6M U , nosTOMy Ff —LF, rae F — i/i, — /Zj,oTKyfla
U — LiFj—LiL F — LiF . Mbi MOJKSM ciHTaTb, MTo L adenGBa.
noarpynna us L odnanaet aononaemieM B
noflrpynne H3 U , v. noaTovry L oSjiaaaeT B U
6M ( d 1 3 . TeopeMa 15.8.6). JleMMa AOKasana.
/1EMMA 4 . E c j i n C / - K O ! : e H H a a n p o c T a s
n a a S e n e s a r p y n n a n o K a a a T e n a 3 0 , T o
n o p a n o K c n j i o B C K - o f i 2 - n o f l r p y n n w 7"
r p y n n w £ 7 - n e T n a a c T e n e H b H H c n a 2 .
HOKASATE/lbCTBO. N (T) y& HMGGT B T nenoflBHMHbix roieK ,
7~ - MHHHMajit>Haa flonycTHMaa OTHOcHTenfaHo J\f(T) nog. -f
rpynna B T , TO J 7^ / =£ 2, . yV= /C(Tt} - HeeflHHHinaH 3,5-rpyn-
na anHe0Hbix npeoSpaaosaHHJl npoorpaacTBa 7^ . IlycTb No - MHHHManbHaa
HopMonbHaa B N" nonrpynna. IlocKOnfaKy -V no TeopeMe EepHcaftna (L\3,
reopeMa 16.8.7) paapeuiUMa, -A^ KOMMyTaTHBHa. FlycTb To - HenpHBoan-
MOe OTHOCHTenbHO HO noAnpocTpaHCTBO B Tf , rorna Tf ecrb npaMaa
cyMMa nojjnpocTpaHCTB, pasMepnocTb Ka»Aoro HS KOTOPHX pasna paaMep-
HOCTH 7 ~o , H flOCTSTOHHO HOKaSaTfa, HTO pa3M9pHOCTb 7" H6THa. Ho 3TO
AeficTBHTenfaHO Tax, nocKonbKy na 7" HenpnsoaHMo fleftcTByer rpynna
0 jC ' (T ) nopaflKa 3 HJIH 5, H paaMepuocTb 7" paDHa 2 HJIH 4.
OcjiaSneHHaa npognevia Bepncaftfla ana noKaaaxeJia 30 465
Ma AOKaaana.
JIEMMA 5 . E c n H U - K O H e n u a f l H e p a a p e -
in n M a a r p y n n a n o i c a a a x e j i a 30 H e c n n
n 10 6 o f t H e p a a p e i i i H M b i J i K O M n o a H u . H O B -
H bl ft <JD a K X O p C/ H 3 0 M O p < } > e H /? _ , T O B Z7 C O -£
a e p j K H x c a n o a r p y n n a H c o c n e a y i o m H -
M H C B O f t C T B a M H :
a) /T' c o a e p J K H X c i i n o B C K y i o 3 - n o a -
r p y n r t y H 3 U ,
6 ) / - / = / 7 x S , r a e / 7 - 3 - r p y n n a , * B ̂ f l f .
AOKASATE/IbCTBO. Ho jieMMe i H S ^ ^ C B . r f l f e C ^ L / ,
O — /7 ,. , £7 ^^5 == ^ . riycTfa /? - cH/ioBCKaa 3—noarpynna Ha C
Toraa HopManHsarop H & O HepaapeiiiHM H conepJKHT cHnoBCKyro 3—noa-
rpynny Ha U . Mbi MOJKBM BMSCTO U paccMOTpexb yV^. (V7) H CHHraTfa,
HT» C7= TVjj t/3) . Torna /-IB -noarpynna, H naM ocxanoc^, noxaaaxb,
<jxo B neHxpajiHsyer ft . FIocKonbKy B - npocxaa rpynna, flocxaxoq-
HO ySeflHXfaca, HXO Cg C&) =£ d . Ilycxfa D - noarpynrra HS B , nao-
Mop<Jmasi fl, • ECJIH Cjj(.A)—-l, xo no xeopeMe B Ha C3J/75 coaepjKHx
aneMenx nopaaKa 9. IloaxoMy .Cn C/?) = 1 , n neMMa flOKaaaaa.
/IEMMA 5 A . E c n n /-/ — K O H e ^ H a a 3 - r p y n -
n a n o K a a a x e n a 3 , o 6 n a a a i o f f l a a n o a -
r p y n n o f t / 7 n o p a n K a 9 , c o B n a n a J o m e f t c o
C B O H M n e n x p a j i H a a x o p o M B / - / , x o / / V / ^ 3 .
flOKA3ATE/IbCTBO. FlycTb B> - MaKCHManfaHaa aSenesa noarpyn-
na us H , HHBapnaHxHaa B H . fl n S> > 'Z (H), H ecna \Z(H)l>3 , xo
fl=H , H neMMa noKaaaHa. ECJIH Z(.H)—$, xo /7 = Z(H) X ^ re>
ECJIH /74-B , xo /7 = £ , n.nocKoJifaKy CT/5; = B ,\H\^-27 . Ec-
nn fl4B , TO x-4 B H ICa(x:)\ lZ(/-/)l=3. nocnontKy H ne co-' &AepKHX sneMQHxoB nopuflica 9, >KOpflflHoaa (JiopMa nnaeiiHoro npeofipaaosa -
HHH •U—+-JC. voc npocxpaucxBa B ae coaepKHT KJIOTOK paaMepHOcxn 3
n nopaaoK B pasen 9. Tan KaK C(&)=SiB , \H\ , H neMMa AO-
Kaaaua.
/1EMMA 8 . E c J i H ^ - K O H e H H a a n p o c x a a
H 0 a 6 e n e B a r p y n n a n o K a s a x e n a 3 0 H
T - e e c n n o B C K a a 2 - n o a r p y n n a , x o / 7 " / ^ , 2
H n H U ^ fl_ .5
flOKASATE/IbCTBO. Ilo npeabmymefi neMine aocxaxonHo
466 B.fl.MaaypoB
HTO B cjiyiae | T\ V U — A 5
rio npennoxeHHio 6 HS [2]
'C^/3 .. (I)rne £ - HHBOJiK>mia HS T , ^ , x-f , JC^ , 3C3 - cxeneHH aenpHsoaHMux
KOMiuieiccHbix xapaKTepOB rnaBHoro 2-6noxa rpynnw £7 , x .• = £„• (mod 4),L- It
t =• ± i H ^ -f- £-« + £ ^ + ^X3 —O.CLr -6f 2 .̂ a 2, .
rpynna H X1 , x^ , x. HeieTHbi, JC.£=3 i> H <2t- ~T~ V£ ^ O . Pa-
B&HCTBO t,^, •+• t,x^r£.3x:,>'= noxasHBaeT, TTO OHHO us
JK6M, ^ , He aenHTCH na 3, a O.HHO HS ^c^ , cKaxceM, J2_ ,
ca na 5. rioaxoMy JC=5'a, x = .3 H JC,= S°3 • Ilycw. BHa^ane x.1 a y j
na 5. Toraa J~*~£2 3 =-O(mods) H 6 - nemoe IHCJTO. IIOCKOJII.—
Ky ^^ = 6^ (mod 4), £z—l • <£r=- i , oricyna £^ = ~1 s. d. Heier -c ^
HO. Ha paseHCTBa ( I ) no/ryiaeM, ^TO 5-3 — 1 ne aenHTca na npoo -C rS
HHcna, OTJiRHHbie OT 2, H ne aenHTca Ha 4. 'IIoaTOMy 5-3 — 1 =-2,.
HTO npoTHBOpenHBo. ^c. He aenHTCfl Ha 5, Torna jc, = 3 , H / + £ =»& • B cL •
= 2 -3 =~(£.3 •'"43 ;l OycTii ana onpeneneHHoc'TH 6 T-cl , roroa
2;? -£3= 2 • ECJIH 6=£d , TO 6-d=l,£3= i, £z=-l. .
^^ H 3 °— i =• 2-5& , rue 5^- o . Ho Toraa t$ - HBTHoe HH-a
cno H 3 — 1 flenHTCH Ha 4. FIosroMy 6=-cL , ~£-2~~£3 = 2 , TO ecTfaa ^f
£,= £. = — i , H HH OflHO H3 1HC6JI 3 — / TS. • 3 ~1 He flenHTCH Ha
5. Ho as paseHCTBa (1) 3 — / nenHTca nnmb na npocTbie qiicna,
mne |C/I , H ne nenHTca na 4. FloaroMy 5 — i — 2 u &„ = az,& 3U - KOHeiHaa npocTaa rpynna o6naaaK>maa HenpHBOAHMbiM
HUM npencraBJieHHeM cTenetfn 3, oricyna U ai /?_ . /leMMa AOKasana.5^
3. MaKCHManbHue sneMeHTu B M(TJ
B 3TOM nyHKxe ^ - KOHeiHaa npocxaa rpynna noKaaaxena 30, H
tj>aKTopbi nroSott co6cTBeHHoft noarpynnw rpynnu 6- na-
60 paapeuiHMw, na6o H30MOp<})Hbi A ' . T osaaiaeT i))HKCRpoBaHHyio canoB-3cKyio 2— nonrpynny rpynnw Cr . Mw noxaweM, HTO 14 (T) o6nanaer eamj-
CTB9HHblM MflKCHManbHUM 3neM6HTOM. Flo J19MM6 6 Mb! MO>K9M CHHTBTb,
\T\t i6 .
/IEMMA 7 . I l y c T f a a n a H e K o r o p o f t H H B O —
OcJiaSneHHaa npoSneMa Bepncafifla ana noKaaarena 30 467
J I K I U H H t- H 3 T u e H T p a J i H s a T o p ^ B r p y n n e
G- H e p a a p e u i H M . T o r A a H e K O T O p t i H M a K—
C H ' M a j I b H b l f i 3 n e M 6 H T H 3 I/I ^ (7~) C O A e p J K H T -
c a B O (C C V j ) , r n e V - H S K o r o p a a n o n r p y n -
na n o p a a K a 4 H a 7" .
AOKASATE/IbCTBO. T KOMMyxaTHBHa , nosroMy Ctt) > T . no
rue A^/7^ (i=i,...,k) . Ho neMMe 3 C(.t)=-*F ,rne F ̂
H XnF=-4. MDJKHO ciHTari., HTO /^ conepwHTca B ueHTpamiaaTOpe CH-
noBcxofi 2-noflrpynnbi 7^ as X . HCHO, HTO 17^,1 ^4 . ECJIH IA '^ (T) He
conep>KHT HeenHHHHHbix noflrpynn, TO neMMa cnpaBeajiHBa. FlycTi. P — He—
eflHHHHHbifi sjieMBHT H3 H « (T) . Torfla B TO cymecTByer HHBoniouHa U
raxaa, HTO C~(.u)^ I .{^CpLu), T 3 HMeeT HeieTHbifi nopaaox. Flo neMMe
2 Cp(ut ^ OCCctJ') , H, rax KBK F ^ C(.u1 , nopssoK C(u) aejiHTca Ha
9. Flo neMMe 5 B C(.u) conepjKHTca nonrpynna ff x B , conepjKamaa CH-
JioBCKyro 3-nonrpynny HS C (.U.) , H fl - 3-rpynna, a. B—#r" Tax K&S.
nopanoK C (u) nejiHTca aa 9, /?^= 1 . flycTb H - nonrpynna MaKCHMajib-
Horo nopanKa co cnenyromHMH cBoficTBaMH: H = C *J) , rae C - 3-rpynna,
a 7} zz- fl £ . IlycTfa H ne coaepJKHT CHJIOBCKOH 3-nonrpynnw rpynnw G- .
ECJIH C Hea6ejieBa, TO C xapaKTepHCTHiecKaa noarpynna B CHJIOB -
CKOfi 3-noarpynne rpynnw H H JV(C') - HepaapeuiHMaa nonrpynna, B
Koropofi nopaaoK CHJioBCKoft 3-noarpynnbi 6onfaine nopanKa CHnoacKofl 3-
nonrpynnbi B H , H no JieMMe 5 nonynaeM npOTHBOpeine c MaKCHManbHO-
CT&K) /-/ . HTBR, C a6eneaa. ECJIH X. - npoHSBonfaHbiH sneMeHT us nop-
MajiHsaxopa cnjiOBCKofi S-noarpynnw rpynnw H B cnjiOBCKOfi 3-noArpynne
rpynnw G , TO jiwSaa >KOpaaHOBa KneTKa jiHHefiHoro npeo6pa3OBaHHa CH-
3-noarpynnw H , cooTBeTCTByroiaero X, , HMeeT paaMepHOCTt He
nsyx, HHane, G- conepxHT sneMeHT nopaaKa 9. ECJIH Tenepfa nopa -
C 6onbme 3, TO X. ueHTpajinsyeT HeKOTOpwft HeeaHHHHHbiH
y HO C , H Cfy) - HepaapeuiHMaa nonrpynna, nopaaoK
3-nonrpynnbi KOTOpoft 6o.rifauie nopanKa CHJIOBCKOH 3-noarpynnbi rpynnti H .
FIosTOMy B CHJiOBCKoft 3-noflrpynne Cr ̂ rpynnbi G cyinecTayeT nonrpyn-
na H 3 nopaAKa 9, cosnanaiomaa co CBOHM uenrpajiHaaTopoM B G3 , H
no neMMe 5A cHnoscKaa 3-noarpynna rpynnw C- Hea6ejieBa nopanxa 27.
HTax, nn6o rpynna t-f conep>KHT cHnoBCKyro 3-nonrpynny rpynnw Cr , JIH-
60 CHJioBcxaa 3-nonrpynna rpynnw G Hea6eneBa nopanKa 27. riycTb, BHa-
CHJioBCKaa 3-nonrpynna rpynnw G HMeeT nopanox 27 H P - MBK -
468 B.fl.MasypOB
sneMeHT B I/I ' ( T) . IIocKOJifaKy iHcno oSpaayroraHx rpynnw 7"
ne Menfaiue nexbipex, a 9 , CT (P.)=CT (
H jieMMa B 3TOM cnyiae noKaaaHa. Dycxt H conepKHtx cmiOBCKyio 3-non-
rpynny rpynnw G- . Flycxt. V - cHJioscKaa 2-nonrpynna rpynnbi H ,
Torna jV" =• -A/^ (V) conep>KHT CHJiOBCKHe 2- H 3-noarpynnw rpynnw G .
ECJIH cunoBCKaa 3-nonrpynna P HS O (N) He paBHa 1, xo Pcl/l3(.T) H
P < C CV) . Ilycxfa P coaep»<nxcH B MaKCHMajifanOM aneMeHTe P us
H ( . T ) v L P ={=. P* . nycxb P^JV+CP) . Toraa N(P) coaepwHx CH-
noBCKHe 2-H S-noarpynnbi rpynnbi & H O(^C(y]}J^ (P}=K. IloaxoMy
3-nonrpynna 0( /P) pasHa 1. C apyroft cxopoabi, /P H&XLKT B
' T/7 H no neMMe 2 /P nescHT B Q(Sf(P)/P) . npoxHBOpeiHe._t y
-^X-7 •& P ^= C(V) . TaKHM oSpaaoM, GCJIH
HOKaaaHa. Ilycxfa f O (Af)l He nennxca na 3.
y
^= C(V) . TaKHM oSpaaoM, GCJIH I 0(N)\ na 3,neM-
H B / 0(jf] CHJioBCKaa 2-nonrpynna HHBapHaHXHa, N(T) conep>KHT
3-nonrpynny JV3 HS W . Flycxb C,— N.. n C, (V ) . Torna•̂ J 3
\ff • C 1 == 3 H V HHBapiiaHTHa B 7"-A/̂ . ECJIH nio6oH sneMeHX HS
Ha 7" 6es HenonBH>KHbix roneK, xo nycxt,,
C3, i/e(C3r>Z ( S f 3 ) ) . (3aMexHM> HTO C3 ^STj H, no-
CKOJifajcy yVCVj nenHTca Ha 9, C,^ 1 ) • Hycxfa Tg - jiroSaa MHHH -
nonycxHMaa oxHocnxenbHO -i oc,y> noarpynna as 7"
-'y neftcxeyex Ha 7"^ 6ea HenonBHMHbix XOHOK, y
ayex TO . riocKonfasy 7" - npaMoe npoHaseaeHHe MHHHMajifaHfaix nonycxH -
MHX oxHOCHxenfaHO -̂ .̂ c, y> -noarpynn, y uenxpaniiayex 7" , HO xoraa no
2 y ne>Knx B nonrpynne 0 (-W(V)) , nopaaoK Koxopofi no yc-
He nejiHxca na 3. Mraic, cymecxeyex X. B /^ ^ ^* • neHTpanHayra-
maft HeKOTOpwfi aneMeHX B T . OocKOJifaKy niofiaa MHHHManfaHaa nonyc -
XHMaa oxHocHxenfaHO 3C nonrpynna B T HMeex nopaaoK 2 H nopanOK
7" no neMMe 4 - nexHafl cxenenfa aBoflKH, I C T ( _ X ) I ^••4 . riosroMy ec-
JIH P - HeenHHHHHbift sneMeHT B I/I* (7~) , xo cymecxByex HHBOniouHa H B
, nan KOTOpoft £„ (U)^ 1 . no neMMe 2
B C(u) nejKHX <i "V, JC > — fl ^ . PaccMoxpHM noo.
ft - paapeuiHMaa nonrpynna. Ilycxfa U - xojinoBCKaa 2,3-nonrpynna H3
fl , conepjKamaa <^ T , x ? . Torna no reopeivie B na C 33 U *"> O (C(U-v
nexKT B uenxpajiaaaxope V , nosxoMy U ^ O(C(u)) HSKHX B O(Af(V)),
HO Un OCCCu)) -oHjioBCKaa 3-nonrpynna aa O(C(U)) , Koxopaa no ycno-
_ Ocjia6ji3HHafl npoSjieMa BepHcaftga ana noitaaaTejia 30 469
BHK> HeeflHHHHHa. HTBK, |£7(W(V))I nejiHTca Ha 3, BonpeKH npennojioxce -
HHKD. /leMMa noicaaaHa.
/IEMMA 8 . F l y c T f a u e H T p a J i n a a T O p b i a c e x
H H B O J I I O U H H K B T p a s p e i i i H M b i . T o r a a a n a
j i r o S o r o H e n e T H o r o n p o c x o r o H H c n a ̂
B e p H o O H H O H a c J i e a y r o i i i H x y T B e p . w : a e -
H H ft:
a) a n a n K> 6 o r o s n e M G H T a A' H a ^/^ U)
c y r a e c T B y e T n o a r p y n n a n o p a a K a 4 u s
/ " . u e H T p a n i i s y r o i i i a a P ,
6) /'/_ ( ' T ' J c o a e p K H T c H j i o B C K y i o / 7 - n o n -
r p y n n y r p y n n b i (j
flOKA3ATEJlbCTBO. ECJIH l/l p (T} coaep>KHT TOJIBKO enHHHHHyra
noarpynny, TO sepHO yrBepwneHHe a) . IlycTfa MaKCHManfaHbifl aneMSHT P& t f p ( T ) ne paseH 1. ECJIH P ne asnaercfl CHJIOBCKOH p - nonrpyn-noB rpynnbi & , TO P He asnaexca CHJIOBCKOH p - nonrpynnoft B
. ECJIH /V(P) - HepaapeuiHMaa nonrpynna, TO no jieMMaM 1 H
3 tf(p}=flP, , roe fl< M (P) , & tz. fl 5 H flnB = i. CHrtOBCKaa
2-nonrpynna TO rpynnw fl no ycjioBHio HeeflHHHHHa H N (TQ) Hepas-
K(.T0}~flE> , rneA*3Sf(To),&^/lf.,fin&=4 , H, no -
Cg CT"O) coaepwHT CHJioscKyio 2-nonrpynny HS B , B-£-C(To) H
B Cr ecTb HHBOJirouHa, uenTpajiHsaTOp Koxopofl HepaspeuiHM. FloaroMy
W CP) - paapeuiHMaa noarpynna. riycTt, U - xowioBCKaa 2, p -noa -
rpynna HS JV(P) . O(U) ^ P H 0(V) = P , xaK KaK Q(C7) ̂ MpCT) .
HocKOJitKy £7 noKasarejia 2z? , no reopeMe B HS [133 B U/O- Cff)cK-
no-BCK.au p -nonrpynna HHBapnaHTHa. ECJIH \ (U)\ , TO P £
H Bee noKaaano. ECJIH I 0^ (.U)\ -4 , TO O. (£7) 4= Z (U) H
p -noarpynna HS £7 HHBapnaHTHa B C7 , TO ecrs P=0(U)- cnnoBCKaa
£7 -noarpynna HS C/ . HTO He TBK. /leMMa noKaaaHa.
/IEMMA 9 . C y r a e c T B y e T ^ ? n a M H O x e c T -
B a { 3 , 5 ^ , a j i a K O T O P O T O B b i n o n H e n o c n e -
a y i o i n e e y T s e p w a e H H e : a n a j i w S o r o s n e -
M e H T a P K B l/l p (_T) H a f i a e T c a n o a r p y n n a n o -
p a f l K a 4 H a T ' . u e H T p a j i H a y i o m a a / 7
flOKASATEJlbCTBO. B cnjiy JIOMM 7 H 8 nocTaTOHHo HCKJiroHHTfa
cjiynaft, noraa flna nio6oro p = 3,5 CHjioscicaa p -noarpynna rpynnbi
(j JI6>KHT B I/I (T) . FlyCTfa 3TOT CJiyHafl HMa»T M6CTO H (jp - CHrtOB-
cKaa p -noarpynna rpynnw G- Ha 14 (T) ana p =3 H 5. Toraa Cr =
470 B.fl.MaaypoB
= G, Gg- T . IlycTb 3C£ N (T) \ . IlycTb U. - HHBOJUOUHS B T Ta-
nas, HTO x, LI oC- = TJ. T= U- . Torna 3d, = Ct. ct r "C , rue C[ £ G^ ,
IlocKonbKy T HopMaJiHayeT G. H GS , Qg ^9V ==<y^s ^ ' a $j ^9$ ~
- 9» ^ . rne q' e G,- , G' £ G . . Mw nonynaeM pasencTBo g.'u =g u ,' •* fS i> 03 J " •> a-3
oTxyoa gi.=-q'u-ii'e. (s}T . riocKonbKy G^T -2,3-rpynna, a Cf5 -5 -/ / faneMeHT, Q'_ = / . TOHHO Tax see c/ = i H U = U- , HTO He Tax. JleM-35 /3
Ma floxaaaHa.
JIEMMA 1 0 . J l n a j i r o S o r o n p o c T o r o H H C -
n a p j i K > 6 b i e 2 s n e M e H T a X 7 H / C / H 3 M? f TV
c o n p a x e H H c n o M o m b i o s n e M e H T a H S
AOKASATE/lbCTBO. Hycrfa neMMa HesepHa. B 7 cyinecTByer HH-
raKaa, ITO Cp(.'t)==C^=- 1 H C„ (.£)=CT^ 1 • Oo neMMe 2
~,C1'?'^: O(C(£)) , H MOWHO HafiTH 3Jl6MeHT OC- B C (.T)
HTO < £* , C ̂ - p -rpynna. HycTb cpeaa Heconpa>KeHHbix c noMOuibro C(7~)
nonrpynn HS / ^ ' C ? ~ ) P v. P BwCpaHH Tax, HTO nepece- x
P.nP — P/ a
Flo npenbiaymeMy aSaauy P n P ^. j
nO J16MM6 2, H MOJKHO Bbl6paTb TJ B C(T) T8K,
y? -rpynna, HTO npoTHBOpenHT MaKCHManbHocTH Pf /"> P. . JleMMa
sana.
/IEMMA II. l/\(T) o S n a a a e T e n H H C T B e H H M M
flOKASATEnbCTBO. HycTb O = OCCCT)) , nycTb O^ - CHJTOB-
CKast 3-noflrpynna, Oc -CHnoBCKaa 5-noarpynna rpynnw O , B^ - Max-
CHManbHwfi SJISMBHT B M,(*) , conep>KaiiiHfii Oi , Bj- — MaKCHManfaHbift
sneMeHT H3 I/I,. (. T) , coflep>Kainnfi: Or- . rjoxasceM, HTO &, &r- -non—y y S 3
rpynna. IlycTb p £. {3,5} ynoBneTBOpaeT saxniOHeHHK) neMMbi 9 u
nycTb V - noarpynna nopanxa 4 HS T , ueHTpajinayioinaa Qp • FlycTb
^ ^ ^eTca CHnoBcxoft ^ -noarpynnoft B £? (^C (a,]) . fleflcTBHTejibHO, no neMMe
2 Cg (.u.) ^= O (C (U)). FlycTb Q_ - canoBcxaa ^ -noarpynna HS O(C(u)),
conepKamaa Ca (U) u newamaa B l/t... (.T) . IlycTb 0 - MaxcHManb -^ ,„ ,^ * „Hbift 3JI6M6HT us A/,, t / / , conep>Karanft ^ . Torna no neMMe 10 cyme -
* -ACCTByex sneMSHT C.& C' C7~) ^~ £, Ci-t) Taxoft, HTO Q = B0 . IlocKOJibxy
^ /-, ^ A cGLnCtu-)- CHnoBCKaa ^ -nonrpynna H3C(iz ; , [Q n C (ul) =
Ocna6neHHafl npoSneMa BepHcaftna ana noxaaaTejia 30 _ 471
=& 9n C (V-) =CB (t/)-cHnoBCKaa a -noflrpynna HS C (t-^) .
TOHHO ran ace B> _ asnaeTcg cujioBCKOft p -noarpynnoft us C
ana UG V , u nosroMy S>p^B (U) = Q(C(.U)) = C
fla BpB>^= Bp<CB (uj \u e V*> =• &^Bp H p
erca noarpynnofi. FloxajKeM, HTO BBn asnaerca ennHcTB6HHbiM MBKCK -
sneMeHTOM B frf (T) . Ana sroro aocxaTOHHo noKaaarB, HTO
^p ̂ o conepxHT jiioSofi MaKCHMantHbift aneMeHT HS A^j CT) u
Ho nioSoil MaKcmvtanfaHbifl sneivieHT HS A^ (.T). u Mg.(T) conpaaoH c
noMOfflfaro aneMenra HS C-CT) (sHaiHT, H c noMomfaio aneMeHxa us
O(C(_T)) ) cooTBercTBeHHo c ^ vunn Bg- , HO B.Bg- 7- O, O^ =
= 0 (.CCT)) , H see
4. Conpa>KeHHOCTi, acex HHBonrouHfi rpynnw G-
KaK H B npeauflymsM nyHKxe, sflect, ^? - KoneiHaa npocraa rpyn-
na noKaaarena 30, T — ee cnnoBCKaa 2-noarpynna. HepaapeuiHMwe KQM-
nosHUHOHHwe 4>aKTOpbi jiK>6oft co6cTBeHHoft nonrpynnbi rpynnbi G H3OMop<J>-
Hbi ft ,. . MH HCKJUonaeM cnynaft G — A ~ , ana nero, nojitayacfa neMMoft
6, flonojiHHTenbHO npennonaraeM, HTO I T] %• 16
B 3TOM nyHKTe MM noKaweM, «JTO / T\ 16 u ITO ace HHBOJIIOUHH
rpynnw & conpHxenbi.
/1EMMA 1 2 . F l y c T t C / - r a K a a c o S c T B e H H a a
n o f l r p y n n a K O H e H H o f i n p o c r o U r p y n n b i
H c a S e j i e B H M H C H H O B C K H M K 2 - n o f l
r p y n n a M H , H T O U c o a e p J K H T H O p M a n n -
s a T O p G H J i O B C K o i t 2 - n o a r p y n n b i r p y n -
n H H H u e H T p a j i H a a T o p a A / n i o 6 o f t H H -
B O J I I O U H H H a C / . T o r n a a c e H H B O J I I O I I H H
r p y n n b i H c o n p a x e H b i .
AOKASATEJlbCTBO. HoicajKeM BHanane, HTO U nepeceKaerca c
OT nee conpaaceHHbiMH c Hefi noarpynnaMH no nonrpynnaM ne-
neTHoro nopaAKa. IlycTfa U - HHsonrouna HS U ^ U , roraa tie. U H
U. € U . B U coaepwHTca sneivieHT ^ raKoft, HTO U. ' w. u.
B oflHoS cnnoBCKOfl 2-noarpynne HS U • FIocKOJUiKy u. H u, i
conpa;*eHbi B rpynne H , OHH conpasceHbi B HopMajinsaxope CHJIOBCKO&
2-noarpynnbi Ha U , B nacTHocTH, conpaxeHw B U ./
HTO u. = u.x ui z H aneMOHT JC u.z£e.C(.U-)^ U
472 B.fl.MaaypoB
H U ==- U . FlycTb renepb U. - npoHsacmtHaa HHBOJUOHHH HS L7 n He
Bee HHBOJIK>IIHH rpynnH H conpsoKSHw. Toraa B H N LI HaftoyTca HS co —
c U. HHBOJIIOIIHH. IlycTb V - oaHa HS HHX. PaccMorpHM sne-
. ECJIH ero nopaaoK HeneTeH, TO U. n V conpaxceHH B N ' (<uVTf),
noaroMy u H u Jieacax B ueHTpajiHsarope HeKOTOpofi HHBOJHOUHH z .
Z£C(L£)^ U H O€.C(Z)^=U , BOnpeKH Bbi6opy V .
JIEMMA 1 3 . E C J I H s e e H H B O J I I O U H H r p y n -
n b i G c o n p a > K e H b i , TO \T \ 16 .
flOKA3ATE/lbCTBO. Ho ycaiOBHio ace HHBOJIIOHHH rpynnti 7" conpa —
»ceHbi B AT T) , noaTOMy \T\ / flennr \Sf(D\ B nacTHocTH, \T\ 1 =
m ?> $ . | T\ 2 . ITycTfa k — 2 • k , rae k. HeneTHo, roraa^ L~2, ' — 4 ne aejiHTca HH Ha 3, HH Ha 5, nosroMy 2.' — / = l . ECJIH
TT.?, $ , TO \T\ 1 nenHTCH na 2 . — 1 = /?• 16 . FIosTOMy 7-1^2 ,
\T\ 16 . H neMMa aoKasana.
JIEMMA 1 4 . I l y c T f a f t - M a K C H M a n f a H b i f t s n e -
M 6 H T B 44(7") . JS(/J) C O A 6 p > K H T /̂V"- ( T) H U 6 H-
T p a j I H S a T O p B G J I K > 6 o f i H H B O n i O a H H H 3
T .flOKASATE/lbCTBO. rio neMMe 11 /7 - enHHCTseuHbifi
B I/ICT) . riycTfa U==--^(^) H nycTi. ac e. M~ ( T) , xorna
fl G. tfCT) H, B CHJiy eflHHCTB6HHOCTH /7, /7 =• /7
T . ECJIH £7 CZ-^ paapeuiHM, TO
Cit) nepaapeuiHM. Toraa
nocTaTOHHo aoKaaarfa, HTO nojiHwfi npooSpaa ^ Y . - . x Li K
C (.t) jiescHT B U , IlycTfa S>2 — S (C (.t)) n T . IIocKojifaKy ~te.
H nopanoK T - Hernaa cTeneHi, iHcna 2, | S, I > -4
AOCTBTOHHO nOKasaTb, HTO £(S~) •£• U . riycTfa i/€-C(S )i/ u y -J Z
Torna ft jiejKHT B I/I (.T ) H, nocKOJibicy 7" %• S, n IS.j f 4-
4: ff
_ Ocna6jieHHaa npo6neMa E«pHcaftna ana nonaaaxena 30 _ 473
JIEMMA IS. I T): = 16
flOKASATE/IbCTBO. npennono*HM, HTO neMMa HesepHa. Torna
\T\ 16 no neMMe 4» Flo neMMaM 14, 13, 12 Mbi MQ>K»I* ciHxaTb.qxo
0(C(t)) — 4 nna niofioft KHBOJIIOIIHR £ € T . ECJIH neHxpaniiaaxop nio-
6oft HHBOJUOHHH H3 T paSpeiUHM, TO .WO") = £7 yflOBJI«XBOpaeX yCJIOBHflM
neMMbi 12 H no neMMe 13 \T\ . IlycTj> cymecxayex HHBonioiiRa t ,
ana Koxopoft C ( £) HepaspeniHM. Flo neMMe 1
rue L^^Ag^—l^k.) . OycTi, X = ̂ C(u)lueT , C Cu) aepaapeviiiM
HTO X ^- J\f (L, ) . IlycTfa u € T H C(U) HepaspemnM. Torna
C (u) =£/ x L2* ...* £s* S(C(u))
T0 =•
Torna \ : TO\ 16 * \TQ\-» 4. C(Tff) * < T,
C(.ro)=Ll*...xixS(CCT0», rae L^ ^ fl ̂ . L, otosnaflaeT c
OHHHM H3 npHMbix MHOKHTenefi SToro paanoxeHHH,
CCTa)-&: SfCL,) ^ L. -^ ^(L,) . tf(T) COCTOHT TOJU.KO HS 1,
)=•!* Jf (L, ) = Lf x...x x X , rfle L^ — fig- , &
— S (JVCL, ) ) ; L>1 cosnaflaeT c oflHHM us MaojKHTeneft sxoro paa-
H
C(u] HepaapeuiHM > - co6cTBeHHaH nonrpynna rpynnu Q- .
)- coScTBeHHaa noarpynna B (j . HCHO, ITO N~(.X) conepwHT J<f(T).
ana nioSofl KHBonioiiHH t O(CCt)) = J, N(T) coaepjKHT ueHxpa-
nnaaTOp nroSoft cBoefi RRBOJIKJUHH £ , ana xoTopoft C(£j paspeuiHM, noa -
TOMy N(X) yaoBneTBopaer ycaioBRjo neMMbi 12 R no neMMe 13 / 77 =s/6 .
J1EMMA 1 8 . B o e H H B O H I O U H H r p y n n b i £ ?
c o n p a w e H b i .
flOKA3ATE^bCTBO. Ho npenbiflymeft neMMe \TI = f 6 . Flycri, neMMa
HesepHa. Flo neMMaM 14 H 12 H (T) COCTOHT TOJIWCO HS 1. EcnR ueurpa -
nRsarop nioSofi RHBOJUOIIHR HS 7" B G paapeuiHM, TO Jf(7~) conep»HT
TpannaaTop nio6oR HHBonrounn HS T , H no neMMe 14 ace HHBO.TCIOIMH
rpynnw G conpaxeHw. FlycTfa t - raxaa HHBonioaHa HS T , HTO
HepaapeuiHM. Flo neMMe 1 C(.t) ^ ff^ *S(C(.t)) H, TBK Kan
C(t) =• £xy _ rfle V - nonrpynna nopaflKa 4 HS 7" , L^-
MOJKHO ciHTaTfa noflrpynnoft naG-L(4,2) . FlosTOMy \^^ /J\T 9.S
R - '/T - a6eneaa rpynna. 3neM6HT nopaflKa 3 vaCct) n JV(T) npea-» / fl O \,MarpRneR / I , me H - HenpRBooRMaa Maxpnua pas -
2, £ - enRRR<waa MaxpRaa. FIoaxoMy / J^(T)f-j~ f HO nenHxca
474 B.fl.MaaypOB
Ha 5 H • r = £. r = V x V , , rfle V^AYTV, Vf <3 JVC T „ B
TOHHO rpH xnacca conpa*eHHbix HHBOJIIOUHH: npencTasHTeneM nepBoro Knac-4f.
ca MOxeT cJiy>KHTjb jiK>6oft sneMear HS V , npeflcraBHTenew Bxoporo -
as V/ , npeacTaBHTenew TpeTt>ero -
ECJIH UGT^ (VUVj) , TO C (u-1 o6jiaaaeT HopManfaHbiM 2-flononne-
H., TBK xaK O(C(ii-))=l, ToC(u)—T. IlycTt ^:e V Hy- j iH-
conpajKeHHaa B G c 3neMeHTOM us Vf H He ne»camaa B
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/I a r e p a r y p a
1. M.XO/in, Teopiia rpynn, M., 1963.
2. R.BRAUER, Some applications of the theory of blocks of charac-ters of finite groups, III, J.Algebra,3 (1966),225-255.
3. 4>.XQ/IJI, r.XHFMAH, p -njiHHa p -paspeniHMHx rpynn H penyic-UHOHHtie xeopeMH ans npofineMM Bepacafiaa, MaTeMaTHxa, c6opHHK nepe -Bonos, 13, N° 2 (1980), 64-104.
4. M.H.CAHOB, PemesHe npo6jieMbi Bepacaiifla ana noKaaaxena 4,XH. aan. JITY, 55(1940), 166-170.
5. A.H.KOCTPHKHH, O npo6neMe BepHcaftaa, MSB. AH.'CCCP, cap.MaTeM. 23 (1959), 3-34.
2. R.BBAUBB, Some applications of the theory of blocks of charac-ters of finite groups, III, J.Algebra,3 (1966),225-255.
6. W.PEIT.J.THOltPSOH, Solvability of groups of odd order,PacificJ.tfath.,13(1963),775-1029.
7* D.GORENSTEIH, On the centralizers of involutions in finitegroups.J.Algebra,II,H2 (1969),243-277.
8.J.G.THOMPSON, Honselvable finite groups all of whose local sub-groups are solvable. Bull.Amer.Math.Soc.,74(1968),383-437
8. J.G.THOMPSON, Honsolvable finite groups all vhose local sub
8. J.G.THOMPSON, Honsolvable finite groups all .of vhose local sub-groups are solvable.Bull.Aner.llath.Soc.,74(1968),383-437
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HocTyniHio 21.7.1869