On locally graded groups with minimality condition for a certain system of nonhypercentral subgroups
We characterize groups without nontrivial perfect sections (in particular, solvable groups) with the minimality condition for the subgroups without hypercentral subgroups of finite index.
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| Date: | 1999 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | Russian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
1999
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/4742 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510908300656640 |
|---|---|
| author | Artemovich, O. D. Артемович, О. Д. Артемович, О. Д. |
| author_facet | Artemovich, O. D. Артемович, О. Д. Артемович, О. Д. |
| author_sort | Artemovich, O. D. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:12:54Z |
| description | We characterize groups without nontrivial perfect sections (in particular, solvable groups) with the minimality condition for the subgroups without hypercentral subgroups of finite index. |
| first_indexed | 2026-03-24T03:04:28Z |
| format | Article |
| fulltext |
K O P O T K I H O B I ~ O M . Y I E H H $ I
Y~K 512. 544
O. ~ . ApTeMOBHq (Hail. yH-'r iM. T. ILIenqeuKO, Kacn)
O Y I O K A Y I b H O C T Y I I E H q _ A T I ~ I X I ' P Y I I I I A X C 3 r C J I O B H E M
M H H H M A d ' I b H O C T H ~YL,q H E K O T O P O I ~ C H C T E M ~ I
H E F H I - I E P I ly~ H T P A ~ ' b I - IBIX H O ~ I " P Y I I H
Wc characterize groups without nontrivial perfect sections (in particular, solvable groups) with
minimality condition for the subgroups which possess no hypercentral subgroups of finite index.
OxapaK'repH3oBani t'pyn• 6e3 tle-rpmda~muHx ItOCKOUaJmx cexiti~ (3oKpeMa, pOaB'aalli rpynH) a
yMol~OlO MilliMa~lhltOC'l'i/tJl:4 niw'pyn, aKi ire MalOTb ~'inep~lewrpaJn, uHx ni/u'pyn cKiHqemloro ill/[eKcy.
l~y~eM roBopHTb, qTO 6eCKoHe'-IHa~q r p y n n a G y~oB~qeTBop~eT yC~OBHIO MHHH-
Ma./IbHOCTH ,~.rLq n o ~ r p y n n , He ~IB.II~IIOII~HXC~! nOqTH FHIIepI.IeHTpaJIbHhIMH
(coKpauleHHO M i n - Z A F ), ec.rlH/Lrl~q ,rllo6ofl y6btBalotueIt rlOCJle/~OBaTe.rlbHOCTH ee
oa py.. { Hnl" N } ae ,Oe t, .TO .oa py..a
H~. noqTa rnneptteHTpa~bHaa ~ g a a ~ o r o s > t. AHa.norr~qHO onpe~e~aeTca yc-
.rloaHe MHHHMa/IbHOffrH ~2I~ HerHneplAeHTpa.rlbnhlx no~rpynn (coKpatttem-io
Min - Z"'A ). IIpHMepaMH rpynn, y~o~e ' rsopaIomHx yc.aoBmo Min - zA, anJ-,amTCa
rpynnu xana XaflHeKeHa - MoxaMeaa (T. e. HeHH2lbrlOTertTHhle rpynnra c Hrl.rlblIO-
TeHTHUMa H Cy6HopMa~bH~aMr~ co6c'r~eHS~Ma no~arpynna~H) [1 - 5], rpynnu qapa -
Ha [6, 7], ZA F-rpynma (T. e. rpynnu, He aB~a~omaeca nOqTH rHnepuewrpa.nr~Hu-
MH, HO C no,~am rHr~epUeHTpa.rmHUMH CO6C'r~eHH~aMn noarpynnaMH) [8].
B /IaHHO~I pa6oTe oxapaKTepHzoBama rpynnra, He HMe~OmHe HeTpHBHa21bHblX
conepmeHHhtX ceKtta~, C yc~Io~HeM Min - ZA F.
Hano~HrU~ HeCKO~.KO Heo6XOtIaM~aX noHsrrai~. Fpynna G Haa~anaeTca aepa3~o-
~KHMO~, ec.aH n~o6~,~e /IBe ee C06CTBeHHbIe no~rpynma nopo:~K/~alOT CO6CTBeHHyIO
no/xrpynny. Ecnn G = G', TO rpynna G Haz~naeTca coneptueHHOia. Be3~e HH~Ke p
rt q ~ paznn,~mae npocT~ae qr~c~qa, k ~ HeoTpntlaTe.rlbHOe tle~oe qncno. Ecru Y
210Ka.rlFHO KOHeqHoe no~e xapaKTepHcTrmn q, a X ~ KaaarlI/~lK.rlHqeCKaYl p-no~-
rpynna r,~yz~,T~n.naKa'raaHOfl rpynnu Y* noza Y, To MHO~eerao nap
G(X, Y,p*) = {(x,y)lxe X, y e Y}
o6paayeT rpynny OTHOcrrre:mHO onepaurm, onpeRe~enno~ no npa~a2 W
(x,y)(u,v) = (xy, xP'v+ y),
~oTopaa Haa~aeTca rpynnolt qaprma (c~. [6, 7]). Hagonett, G - - HM*-rpynna,
ecart ee gOMMy'raTop G' raneptleHTpa.nhma~, a tl~agTop-rpynna G/G" ~ ~earlMaa
'~eprIrmOncgaa p-rpynrm. Oqena~no, wro rpynrt~a Tana Xa.qaeKeaa- MoxaMe~aa n
rpynnr~ qaprma a~a~OTCa HM*-rpynnaMrL
�9 O. ~[. APTEMOBHq. 1999
ISSN 0041-6053, YKp. ,~tam. w.ypn.,1999, m .51 .N e 10 1425
1426 O../1. APTEMOBHq
B pa6oTe/loKa3a~l c~eAy~ott~e ABe Teopcbi~/.
T e o p e H a 1. llycmb G ~ nezunep,enmpanbnaz HM*-epynna.
1. Ecnu G ~ p - e p y n n a , mo G y0oonemeopxem ycnoou~o M i n - Z A F moe-
cga u mon~,xo moeOa, KoeOa G ~ epynna c nop~tanu3amopn~t~t ycAooue~t;
2. Ecnu G ne .~on.~emc~ p-epynnoa, m o G y(9oonemoop~em ycnoeu~o
Min - Z A F moeOa u monbKo moeOa, KoeOa
G = (A>~(S~•215215 k >. 1,
zcge A ~ eunep~enmpaAbnaz q-zpynna, S i ~ tzoa3u~uKnut~ec~ca~ p-epynna, p u
q ~ pa3~ut~m,~e npocmbte ,~ucna, S ~ cge.au~taa ,~epnutcooc~a~ p - epynna, S i
Oeacmeyem mpueuam,no na noOepynne @pammunu 0 ( A ) u nenpueoOu~to na
dpamnop-epynne A Idp (A ), Z (A ) = A" = O(A ) u, rpo~,e moeo, (A >~ S i ) / O ( A ) ,
i = 1 ..... k, ~ epynna tlapuHa.
Teoper, la 2. l lycmb G ~ zpynna, ne u~te~oulaa nempuouanbm,~x cooeputenn~tx
cet<~ua ( o qacmnocmu, pa3petuuztaa zpynna). ToeOa G yOoo,~emoopaem ycnoouto
Min - Z A F, ecnu u monb~O ec.~u G npunaOne~rum oOno~o' u3 munoo:
1) G ~ noqmu zunept4enmpam,naa zpynna;
2) G ~ ~tunu~tanbnaa neeunep~enmpanbnaa zpynna;
3) G = M >~ Q, Q ~ Koa3u~utcauqect<aa p-zpynna, M - - zuneptlenmpaAbna~
q-zpynna, Q Oe~cmoyem mpuouanbno na noOepynne ~pammunu ~ ( M ) u
nenpuooOu~to na dparmop-zpynne MId,(M), Z (M) = M" = Z(M) , u, rpo~te mozo,
G IO(M) ~ zpynna ttapuna;
4) G coOep~um mah3'~o nop~tam, nyto noOzpy.nny D ~one,~noeo unOet~ca, ,~mo
D = D I.....D t, t > 1,
eae D i - - G-Oonycmu~ta~ HM'-noazpynna, npunaane~au4a~ muny 2 u.au 3,
npu,~e~t D'i = D', i = 1 ..... t, u ecnu s ~ k, mo nepece,tenue ~ (Dk /D ' ) N
N n (DslD') nycmoe.
B pa6oTe C . o6oanaqaeT KBa3rlRrlK0"IHqecKylO p - r p y n n y , (7" - - KOMHyTarlT
P
rpyrmta G, Z ( G ) - -uerrrp rpynma G, ~ ( H ) --HHOXCeCTaO Bcex npocT~x ~enwre-
Jle~ nop~IRKOB 3JICMeHTOB FlepHo/~HqCCKOlt rpynma n . Bce ocra~bmae Heo6xo~HMI~Ie
o6o3HaqeHrta a onpeRe.~eHaa co/Iep~aTc.q s [9 - 11 ].
1. ~ a AoKa3aTe~cTBa Teopebi 1 H 2 Heo6xoAabim c~e~y~mrie yTBep~KAeHHa.
JleHHa 1. Flycmb ~pynna G yOoonemsopaem ycnooulo Min- Z A F, H - e e
noaepynna. Toeaa:
1) H y~oonemoop~em ycnooun9 Min - ZA F;
2) ecnu H nopztanbna 8 G , mo dpaKmop-epynna G IH yaoonemsop.~em
yc.ao6uto Min - Z A F;
3) ecnu H -nop~tanbna~ nocgzpynna G, ne u~tetou~a~ eunep~enmpanbn~x
noazpynn Kone,otozo unae~ca, mo dpamnop-zpynna G I H ,tepnuKoecxa~.
~OKa3aTeJ'IbGTBO .Tlebibil:,,l HCCJIO~W.HO H blbI ero onyczaebi.
JIeHMa 2. [ ~ y c m b G - - epynna, ne u~tetou~a., nempuouanb~tx coeeptuennNx
cex,ua u ne ~en.~ou~a~ca no,~mu zunep~enmpam, nofl. Ecnu G yOoo/~emeopaem
y c n o o u t o M i n - Z A F u ece codcmaenn~e nop~ant, n~e noOzpynma no,~mu
zunept~enmpaa~e, mo G ~ Z A F-epynna.
/~7[o~a3amenbcm~o. I IoczoJn ,zy ~aKTop-rpynna GIG" nepaa.rlo~KHbia, TO oua
Ksa3rIRHKJWtqecza.r Kpobic ~v0ro, zom~rrarrr G' ranepuerrrpa~mmatl. H Tax KaK
G y~toBJ]evsopzer ycJmsmo Min- ZAF, TO G nbiecT cy6nop~ta.nsny~o ZAF-
ISSN 004 / ,~05 3. YKp, ~um. ~'ypn. . ! 999, m. 5 i . i~: ! 0
O JIOKAflbHO CTYHEHqATI:)IX FPYrirIAX C YC~OBHEM MHHHMA~bHOCTH ... 1427
no~trpynny H, npuqeM G = G'H. E c a a no~trpynna H CO6CTaeHHaa, TO OHa
coaepmHTCa a C06CTaeHHOti HOpMa~nbaolt no~rpynne F, u Tor/Ia G = G'F, a aTO
eeaoaMoa~no. IIoaToMy G = H. JIeMMa ~oKa3aHa.
2IeHHa 3. Hycrab G ~ H M*-epynna c rea3u~ugnu*~ecxoa cibarmop-zpynnoa
GIG' . Ec.au G yao6nemeopaemycnoouto M i n - Z A F , mo G - - Z A F - z p y n n a .
] IoKa3amenbcm~)o . KaK H sbltue, G coz/epmnT cy6nopHanbay io Z A F -
nozlrpynny H H, cJle~oBaTeylbHO, G = G'H. H rIOCKOJIbKy G He HMeeT HeTpHBH-
aSmHHX cosepmeHHHX ceKum;l, TO H = G. d'IeMMa/~oKaaaHa.
dIeHMa 4. Hycmb G ~ neeunep~enrapanbnaa gpynna c nop;~ta.abnOa gunep~en-
mpa.abnoa noOzpynnoa A u p-r~asut~ur.au~tectcoa dpam'nop-zpynnoa G / A . Ec.au
G yaoo.aemeopaem yc.aosuto M i n - Z A F , m o G = A .T , eOe T ~ x a p a K -
mepucmu,cecraa Z A F - n o a z p y n n a , dparmop-zpynna G /G' , . t epnurosc raa u,
Kpozte moeo, G u.~teem marylo noOzpynny D Kone~noeo unOetcca, ,r D = B . ?",
npu,ce~t D" = T" u B ~ ae.au.ataa ,r p-zpynna.
,~oga3ame.abcm6o. Fpynna G co/~ep~rlT cy6aopMa.asHym Z A F - n o l x r p y n n y
T a nycTb
T <- G 1 <-- ... <- G k = G (*)
cy6nopMa.nbHUtt pmx, coe~nHammH~ T c rpynno~t G. T o r ~ a / I ~ a ~ 6 o r o a:Ie-
MeHTa g ~ G 2 nozwpynna T ~ HopMa~bna S G t n, KpoMe Toro,
( T g A / A ) . ( T A / A ) = ( T T g A ) / A = C , , p
a 3naqrlT, T g = T H T HopMa.qlbHa B G 2. P a c c y ~ a a ana21orHqHo, qepe3 KOHeqHOe
qnc~o maroB no~yqaeM, wro T HopMa.rlhHa BO BCCI~ rpynne G. . r l e rKo y6e/XHTbC.%
qTO T xapaKTepucTnqecKaa B G.
l-lycTb D / T - - n o J m a J ~ qaCTb qbaKTop-rpynma G/T. HOCKOSmKy qbaKTop-rpyn-
na G / T qepHHKOBCKaJt, TO D HMeCT KOHCqHhlI~ HH~eKC B G. KpoHc TOrO,
( D I T ' ) I ( T I T ' ) = D I T
~e~nMaz qeprlnKOBCKaJt vpyrma H BCa-le/~CTBHe TeopeH~ 1. 16 [9, c. 67 ] cl0aKTO p-
rpyrma D / T ' a6esmBa, a CYIe/~OBaTeYIlbI-IO, D ' = T'. OTClO/~a TaK~Ke cJIe~yeT, qTO
qbamTop-rpynna G / G ' qepHl4KOaCma$1. HaKOHera, ec.rtn B ~ no~Ha~ "tacT~
no~rpynnr~ A ~ D, TO, OqeBH~HO, D = B T, rI HmaZlercz TaKa$l ~CJH4bla.,q qCpHtfKOB-
cgaa nozwpynna C < B, qTO D = CT. 2IeMMa ~aoKazaHa.
d-IeH~la 5. I lycmb G ~nezunep t4enmpa ,qbnaa HM*-epynna. E c n u G yOo(~-
,qemoopaem yc,~oou~o Min - Z A F, TO
G = TI . . . . .Ts .D, s >- 1,
eOe D ~ nop~ta~bnaa 3eau~taa ~epnuKoecra.~ noOepynna, T~ ~ nop~ta~bnaa
Z A F-noOepynna, npu~e~t G" = Ti', i = 1 . . . . . s.
]Ior, a .~meabcmeo . Ho onpe~eaeHmo,
GIG" = N l X . . . x N t ,
r~;e N i, i = 1 . . . . . t, ~ KBa3HtlHKJIHqeCKa..'I p- rpyrma. IlycT~ N i ~ nosmul l npo-
o6paa N i B rpynne G. l 'IocKoabzy G HerlfflepUeHTpaYmHa.'l, TO na~I 'C . , q TaKOC
t~esme ~mc~o s, 1 < s < t, qTO IlO~rpyrllIh[ N 1, .. . . N, HC $IB2LIIIOTCJi HOqTH rHIICp-
UeHTpaabmaMH, a n o ~ r p y n n u N,+~ . . . . . N t nOqTH rrmepuerrrpam, HUe. CorRacHo
ISSN 0041-6053. Yxp, :~uJra. acy. pn.. 1999;m. 51 1~ 10
1428 O./l. AFrI'EMOBHq
neMMe 4, Nj = Tj.Aj, r~r ~ - - xapaKTep.cm.ecKaH Z A F - n o ~ r p y n n a Nj H Tj' <-
<_ N<_ Aj (a~ecl, H HH~Ke j = I ..... S). H TaK KaK qbaKTop-rpy[ma G / T j ~c.rmHa~
'-IepH14KOBCKa~, TO ac~zencTaHe TeopcMbz I. 16 [9, C. 67] c~aKTop-rpyrma G /Tj' a6c-
.rlCBa H, C.TIC~OBaTC.T[bHO, a ' = Tjt. TaKHM o6pa3oM, Aj HopHaJIbHa B G. l ' lycTb
F l = A I....,A s. N,+l. . . . .N t, Ecn14 F - - xapaKTepHcam~ecKaH rHnept~eHTpa~abnaJ~
no~rpynna F 1 Koneqaoro nnaeKca, a B - - aezn~xaa qaCTb F, TO, oqeaH~tno, G =
= T l . . . . .T s. B, s >-. 1, a OTC~aa ~erKo cze~yeT yTBepac~enne zeMMbL
d-IeM~m 6 . H y c m b G ~ neguneptlenmpa.abna,~ HM*-zpynna. E c . a u G
yOoo.aemoop.qem yc.aosuto Min - Z A F, mo
1) r,o~t~tymanm G" ~ q-zpynna a ~ ner.omopoeo npocmoeo ,~uc.aa q;
2) ecAu p ~ ~ ( G ' ) n n (GIG') , mo G ~ p - z p y n n a .
JIeMMa 7. ,/La~ p-epynn~ G, .~oa.~oule~cn H M*-~pynno~, cneSytou4ue ym-
~ep~Seuu~ pa~nocunbnb~:
1) G ycgoanemoop,~em yc.aooua9 Min - Z A ;
2) G ycgoa.aemoop.aem yc.aoowo Min - ZA F;
3) G ~ zpynna c nop~ta.au3amopnbtzt ycnoaue~t.
,O~oxa3ameabcm~o. HMnznKar~na 1) =~ 2) oaeBnatta, a nMnzaKattrsa 3) :=~ 1)
c:~ezlyeT n3 [12]. :~OKa~eM, qTO 2) :=~ 3). I-[yCT~ G ~ aeranept~eaTpa:mna_a HM*-
p - r p y n n a c yCaOBaer~ Min - Z A F, K ~ a~6aH ee no~trpynna. Ec~r~ no~trpynna
G'K runeptteaTpa.n~Haa, TO K ~ CO6CTBeHHaH no/~rpynna cBoero Hop~aanaaTopa
N G ( K ) . l ' lo~To~y npe~no~.oac14M, wro n o a r p y n n a G'K neranep t t e r~Tpa~Haa .
To r~a K coaepacaT n o a r p y n n y S c ae~aMot4 aepHa~OBCKO~ qbaKTop-rpynnoa
G'SIG', 14 BCZe;tCTBae n e b u l a 4 G'S coaepacrZT G - ~ o a y c T a M y ~ o Z A F -
nonrpynny M. B c a z y TeopeMta 1. 16 [9, c. 67] qbaKTop-rpynna G/M" a6ezeBa, a
3na~nT, G" = M'. l 'lprmnMas BO BnaMa~ae zeMMy 1. 1. 1 [13], no~ay~aeM M = (M fq
iq G ' ) (M fq K), oT~y;aa Bcze~tcrBaeTeope~ 3 . 4 n 3 .5 i43 [8] M < K. 3HaqnT, K
aopMa.rm14a B G. ~eM~a aoKaaaaa.
d ' I e ~ a 8. Hycmb G ~ HM*- epynna, ycgoonem~p.~toula.~ yc.ao8u~o
Min- Z A F. Ecnu K ~ noc~zpynna G, m o K no,~mu eunep~cenmpanbna,~ nu6o
G ' < K .
] lo tcazamenbcmeo, lrlycT~ K ~ ~m6aa no~trpynna G, He 14Mezomaz r14,
nep t t ewrpa~max no~trpynn Koueq14oro 1414zte~ca. Tor~a K co~ep~crrr cy6r topMa~-
14y~o ZAF-nolarpyrmy T 14, cornacno zeM~e 4, G ' T = M.D, r a e M ~ xapaKTe-
p14craaecKaa Z A F - n o a r p y n n a , a D ~ Hopuaa~naa r14neptte14Tpa~14aa no~rpynna
G'T. I - I o c K o ~ K y G I M aea14~az qepHaKOnCKaZ, TO, Ka~ 14 B~tUe, G " = M ' 14
Bcze~;CT~14e aeM~ta I. I. I [13] M = T r~ G ' < K . J'IeM~a aoKa3ana.
2. ,~otcasamenbem~o m e o p e . ~ 1. u 1 Teope~ ta c:~e/~yeT 14a
~eMr~a 7. ~ o x a ~ e ~ BT0poe yT~epacaemze. I lpennoaoacH~, aTo G ae HB~aeTCH
p - r p y n n o ~ 14 yao~eTBOpZeT yC~OB14~ Min - ZA F. BB14~y c:xe~cTa14a 6, ~ e ~ 5,
Teopema 3 .5 [8] tz npet~zoacermz 2. 4 [9, c. I01 ] noayaaeM
G = (G" x~ (S~ x . . . x S ~ ) ) x D , k >_ 1,
14 G 14~teeT cBoRcrna, yKaaau14~c n yTBepacaeam! 2 Teope~ra.
Hao6opoT, rtyeTb { K , In r N } ~ K a z a n - : m 6 o y6taBalomaa noc~caOBaTe~b-
HOer~ no~rpynn rpynma G. E c ~ ~.na a m 6 o r o ue~oro n no~rpynna K , He r~leeT
rrmepttewrpaa~HtaX nozwpynn xoHeaHoro HH~exca, TO, na~HHas c HeXOTOpOro u e ~ o -
ro m, HML~M
ISSN 004 ! -6053. Y~:p. ~tum, ~'ypu,, ! 999, m. 51, N el 0
O YIOKATIbHO CTYFIEHqATbIX FPYFIFIAX C YCTIOBHEM MHHHMA.FIIaHOCTH ... 1429
KING" = Km+IG' = ....
Horl.qTHO, qTO K m = (G" ~ Km)>~ L, r~e L ~ 6ecKoneqHa.q qepHm~OsCKa.qp-rpyn-
ha. HycT~ X ~ TaKa~ KBaammKnHqecKa~ no/Irpyrma Ha L, qTO no~rpynrm H =
= (G' n Kin) >~ X He HMeeT ranepCeHTpanhHmX no~rpynn KoHeqHoro ~H~eKca.
Tor~a ~a yCnOBH~ TeopeMm H BBH~y Teol~m~ 3.5 [8] no~aeM ( G" N Km)Z (Y) =
= G', r~e Y= G' >~ X, H, Taxn~i o6paaoM, G" = [G, X] = [G' N Kin, X]< G" N Kin,
qTO HeBO3MO~'~-IO. ~I~O 03HaqaCT, qTO K s IIOqTH FHIIepI~CHTps163 ~,.rl.q HeKOTOpO-
FO IIOJIO)KHTeJIbHOFO L~eJIOFO S. TeopeMa/ioi(aaaHa.
Bonpoc o cyu~eCTBOBaHrlrl COBeptllenublX ~ - r p y n n OTKpUT. HeCJIOYKHO l~O-
Ka3aTb, HTO coBepuleHHaJ/ Z A F - r p y n n a (eCIIH OHa cymCCTByeT) ZB~nZeTCa CqCT-
HOI~I .rlOKaJlbHO KoHeqHOI~I p - r p y n n o ~ , Bce CO6CTBeHHble nonrpynn~, KOTOpO~ rnnep-
tteHTpanbH~ae.
~oxa~ ame nbc moo meope~,~ 2. Heo6xoc~u~tocmb. Flycr~ r p y n n a G He co~ep-
~KHT rnnept~eHTpa~bHmX no~arpynn KOHeqHoro nH/~eKca H yROB-rIeTBop~IeT yCnOBrlIO
TCOpCMbL Ecnn B G Bce C06CTBeHH~C HOpManbHbIe noz~rpynn~ n o ~ r n rnncptteHT-
pa.rlbHHe,'TO BC.rle/ICTBHe IIeMM 2 a 7, TeOpeM3.4 a 3. 5 n3 [8] G - - r p y n n a Tnnoa 2
n ~ a 3. 1-IoaTOMy npe~nono~KaM, qTO G n~eeT co6cT~eaay~o Hop~a~sHym noa-
rpynny , KOTOpa.q He .qBYl.qeTc.q noqTri rrmepUeHTpa0a~HOR. Ho Tor~aa G co~epmnT
cy6HopMa~SHy~O . . Z ~ - n o n r p y n n y T n HMeeT cy6nopManbH~a~t p a a (*), c o e t m -
Ha~Omn~ T c r p y n n o ~ G. Ecnn gmKaKO~-~IrI6OaYteMeHTrl3 G 2, TO T g ' ~ G 1 H
T" = ( T ' ) g = (T#)" ~ q- rpynna ~nJt HeKOTOpOro npocToro q. FIocKOnbKy, cornacHo
~evl~e 1, dpaKTop-rpynna G I / T qepHriKOaCKaJ~, TO G l co~ep~.rrr G2-z~onyerrl-
~ty~o HM*-no~rpyrIny H2. A n a z o r u q n o H~ n o p M a ~ n a a G 2 ~ n z n ~ 6 o r o
~neMenTa a )13 G 3 H T ' = Hi = ( H : ) a = ( H ~ ) ' . H Ta~ KaK qbaKTop-rpynna
G 2 / H I qepnnKOaCKa~, TO G 2 co~epmHT G a - / t o n y c r a M y ~ HM*-no~trpynny H 2.
P a c c y ~ a a noItO6H~M o6paaoM, qepe3 KOHeqHoe qnCnO m a r o a n o n y q a e M
Hop~ansHy~o ~ G HM*- u o a r p y n a y H. Tenep~ Hecno>KHO aoKaaaT~, qTO G r~MeeT
TaKym HOp~4a~ny~o n o ~ r p y n n y KOHeqHOrO nHZteKca, qTO D = D~.. . . 'D t, t > 1 ,
r~e D i ~ xapaKTepHcTHqecKaJ~ HM*-no~rpynna, y~Io~eT~opa~oI~az y c n o a H ~
Min- Z A F . K p o ~ e a T o r o , ecml k~: s, TO n ( D k / D ' ) ~ ~ ( D s l D ' ) = ~ ) , 1 < k , s < t.
I[ocmamo~nocmb. 1-IycTs { K,,ln ~ N } - - KaKa.~-mt6o y6uaa~otua.a n o c n e z o -
aaTeZSHOCTb no/wpynn r p y n n u G n G ~ r p y n n a Trma 4. Ecnn ~t~J~ Kam~oro
u e n o r o n n o a r p y n n a E,, = K~ N D He HMeer rnnepueHTpazhH~X n o a r p y n n KOHeq-
Horo nHae~:ca. TO, HaqUHaZ C HeKOTOpOro u e n o r o m,
E,,,D" = Em+ID" = . . . .
CornacHo ycnoaHm, D" ~ r~nept~enTpan~Haa p - r p y u n a ~nz HeKOTOporo npocToro
p. Pacc~oTpHM BO3MO~Hble cJly,-laH.
1. Ec~n E m COZtep~KnT TaKy~ Kaaa~un~naqecKymq-no /xrpynny X ( p ~ q ) ,
q'ro H = ( E m ~ D ' ) >~ X He rtMeeT rrlrlepUcHTpaYmHblX no~rpynn KOHeqHOrO HH/IeK-
ca, TO a c a n y TeopeM~a 3. 5 [8] nMeeM (E, , N D ' ) Z ( Y ) = D' , r a e Y = D ' >~ X, u,
CIIe/~OBaTeIIbHO,
D" = [O', X] = [O" N E,,,, X] <_ O' t3 Era,
qTO HeBOZMOZC, HO.
2. Hpe~tnonoTxrtM~ wro E, nD" = S x L, r~Ie L - - FHIleptleHTpaYlbHa~ p" -- rpynna ,
I$SN 0041-6053. Yrp. ~tam. ~.'Vlm.. 1999. m. 51. N ~ 10
1430 O. ]~. A P T E M O B H q
a S - - p - r p y n n a , He HMClOtUa.q rHncpttcwrpa~bmax noIIrpynn KOHCqHOrO HH~CKCa.
H e TOt/In, HanpHMcp, D l ~ p - r p y n n a H S -< D I. B cn~y ~CMM~ 5 n TcopcM~ 1 D l
�9 CO]Icp)KHT G-]IonyCTHMyK)Hcpa3JIO)KHMTIO n o ~ r p y n n y M n M ' = D ' . OTClO~a
nBHlly ~ICMMI~ 1. I . 1 [13] D" < M < E m, qTO HCaO3MO)KHO. ~ r o o a u a q a e T , qTO K s
HOMTH rHIIepIIcHTpaJlbHa~I, HatIHHa,q C HCKOTOpOFO IIOJIO;KHTeJIbHOFO I.[cJ-IOrO s .
T c o p c M a ~IoKaaaHa.
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noc..Jte l/opa60"rgH ~ 04. 05.98
ISSN 0041-6053. YKp. ~gtra. ~r pn., ! 999, m. 51, N ~10
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| id | umjimathkievua-article-4742 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:04:28Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/47/6525501b2c6ce303bfc1a2232135b647.pdf |
| spelling | umjimathkievua-article-47422020-03-18T21:12:54Z On locally graded groups with minimality condition for a certain system of nonhypercentral subgroups О локально ступенчатых группах с условием минимальности для некоторой системы негиперцентральных подгрупп Artemovich, O. D. Артемович, О. Д. Артемович, О. Д. We characterize groups without nontrivial perfect sections (in particular, solvable groups) with the minimality condition for the subgroups without hypercentral subgroups of finite index. Охарактеризовані групи без нетривіальних досконалих секцій (зокрема, розв'язні групи) з умовою мінімальності для підгруп, які не мають гіперцентральних підгруп скінченного індексу. Institute of Mathematics, NAS of Ukraine 1999-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4742 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 10 (1999); 1425–1430 Український математичний журнал; Том 51 № 10 (1999); 1425–1430 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4742/6185 https://umj.imath.kiev.ua/index.php/umj/article/view/4742/6186 Copyright (c) 1999 Artemovich O. D. |
| spellingShingle | Artemovich, O. D. Артемович, О. Д. Артемович, О. Д. On locally graded groups with minimality condition for a certain system of nonhypercentral subgroups |
| title | On locally graded groups with minimality condition for a certain system of nonhypercentral subgroups |
| title_alt | О локально ступенчатых группах с условием минимальности для некоторой системы негиперцентральных подгрупп |
| title_full | On locally graded groups with minimality condition for a certain system of nonhypercentral subgroups |
| title_fullStr | On locally graded groups with minimality condition for a certain system of nonhypercentral subgroups |
| title_full_unstemmed | On locally graded groups with minimality condition for a certain system of nonhypercentral subgroups |
| title_short | On locally graded groups with minimality condition for a certain system of nonhypercentral subgroups |
| title_sort | on locally graded groups with minimality condition for a certain system of nonhypercentral subgroups |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4742 |
| work_keys_str_mv | AT artemovichod onlocallygradedgroupswithminimalityconditionforacertainsystemofnonhypercentralsubgroups AT artemovičod onlocallygradedgroupswithminimalityconditionforacertainsystemofnonhypercentralsubgroups AT artemovičod onlocallygradedgroupswithminimalityconditionforacertainsystemofnonhypercentralsubgroups AT artemovichod olokalʹnostupenčatyhgruppahsusloviemminimalʹnostidlânekotorojsistemynegipercentralʹnyhpodgrupp AT artemovičod olokalʹnostupenčatyhgruppahsusloviemminimalʹnostidlânekotorojsistemynegipercentralʹnyhpodgrupp AT artemovičod olokalʹnostupenčatyhgruppahsusloviemminimalʹnostidlânekotorojsistemynegipercentralʹnyhpodgrupp |