Locally graded groups with normal nonmetacyclic subgroups
We establish the solvability of locally graded groups with normal nonmetacyclic subgroups and prove that the degree of solvability does not exceed 4.
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| Дата: | 2007 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2007
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3296 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509358733918208 |
|---|---|
| author | Kovalenko, V. I. Коваленко, В. І. |
| author_facet | Kovalenko, V. I. Коваленко, В. І. |
| author_sort | Kovalenko, V. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:50:22Z |
| description | We establish the solvability of locally graded groups with normal nonmetacyclic subgroups and prove that the degree of solvability does not exceed 4. |
| first_indexed | 2026-03-24T02:39:50Z |
| format | Article |
| fulltext |
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UDK 512.54
V. I. Kovalenko (Çernihiv. ped. un-t)
LOKAL|NO STUPINÇASTI HRUPY Z NORMAL|NYMY
NEMETACYKLIÇNYMY PIDHRUPAMY
The solvability of locally graded groups with normal nonmetacyclic subgroups is proved. It is known
that the degree of solvability does not exceed the number 4.
Ustanovlena razreßymost\ lokal\no stupençat¥x hrupp s normal\n¥my nemetacyklyçeskymy
podhruppamy y otmeçeno, çto stupen\ razreßymosty ne prev¥ßaet çysla 4.
U robotax [1 – 3] vyvçalys\ skinçenni hrupy z normal\nymy nemetacykliçnymy
pidhrupamy. U danij statti vyvçagt\sq lokal\no stupinçasti hrupy, u qkyx
koΩna nemetacykliçna pidhrupa [ normal\nog. Umova normal\nosti nemetacyk-
liçnyx pidhrup [ odnym iz pryrodnyx uzahal\nen\ umovy normal\nosti vsix
pidhrup, wo pryvodyt\ do dedekindovyx hrup. Umova lokal\no] stupinçastosti
— odna z najbil\ß slabkyx umov skinçennosti v zahal\nij teori] hrup. Klas
lokal\no stupinçastyx hrup dostatn\o ßyrokyj i mistyt\ lokal\no skinçenni
hrupy, lokal\no rozv’qzni hrupy, klasy hrup Kuroßa – Çernikova ta in.
Osnovnym rezul\tatom statti [ teorema, v qkij vstanovlg[t\sq rozv’qznist\
lokal\no stupinçastyx hrup iz normal\nymy nemetacykliçnymy pidhrupamy i
zaznaça[t\sq, wo stupin\ rozv’qznosti takyx hrup ne perevywu[ çysla 4. Dlq
dovedennq dano] teoremy vykorystano rezul\tat avtora pro te, wo neskinçenna
lokal\no stupinçasta hrupa z vlasnymy metacykliçnymy pidhrupamy [ abo me-
tacykliçnog, abo kvazicykliçnog hrupog (u podal\ßomu — teorema 1) [4, s. 54;
5]. Teorema 1 ma[ i samostijne znaçennq.
Oznaçennq 1 [6, s. 236]. Hrupu G budemo nazyvaty lokal\no stupinças-
tog, qkwo bud\-qka ]] neodynyçna skinçennoporodΩena pidhrupa ma[ vlasnu pid-
hrupu skinçennoho indeksu.
Oznaçennq 2. Nemetacykliçnu hrupu G iz metacykliçnymy vlasnymy pid-
hrupamy nazyva[mo minimal\nog nemetacykliçnog hrupog.
Teorema 1. Lokal\no stupinçasti hrupy G z normal\nymy nemetacykliç-
nymy pidhrupamy rozv’qzni, pryçomu stupin\ rozv’qznosti ne perevywu[ çysla ço-
tyry. Nil\potentni hrupy takoho rodu magt\ skinçennyj komutant.
Dovedennq. TverdΩennq teoremy [ oçevydnym, qkwo G — abeleva hrupa.
Tomu v podal\ßomu budemo vvaΩaty, wo G — neabeleva hrupa. Nexaj M — pe-
retyn usix nemetacykliçnyx pidhrup X iz G. Za lemog 2 [1] X normal\na v G,
G / M — dedekindova hrupa, vsi vlasni pidhrupy iz M — metacykliçni. Oçevydno,
wo komutant G / M — elementarna abeleva 2-hrupa. Z c\oho vyplyva[, wo G′′ ≤
≤ M, otΩe, G′′′ ≤ M ′, G(
iv
) ≤ M ′′.
Qkwo M ′′ = 1, to G(
iv
) = 1. Qkwo M ′′ ≠ 1, to M moΩe buty lyße skinçen-
nog minimal\nog nemetacykliçnog hrupog (za teoremog 1). Z opysu nil\po-
tentnyx minimal\nyx nemetacykliçnyx hrup roboty [7] vyplyva[, wo M — hrupa
typu 7 teoremy 2.5.2 [8]. Tomu M = P l Q, P — hrupa kvaternioniv, Q — nenor-
mal\na cykliçna sylovs\ka 3-pidhrupa v M, [ P, Q ] = P, P normal\na v G. Za
lemog Fratini [9, s. 157] G = M ⋅⋅⋅⋅ D = P ⋅⋅⋅⋅ D, de D = NG ( Q ), P ∩ D = NP ( Q ) =
= Φ ( P ) i [ P, D ] = [ P, Q ] = P. Qkwo D normal\na v G, to NM ( Q ) normal\na v
© V. I. KOVALENKO, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1 133
134 V. I. KOVALENKO
M. Tomu Q normal\na v M. Pryjßly do supereçnosti. Takym çynom, D ne-
normal\na v G, a otΩe, metacykliçna. Za rezul\tatamy [10, s. 442] G′ = P′ ⋅⋅⋅⋅ D′ ⋅⋅⋅⋅
⋅⋅⋅⋅ [ P, D ] = P ⋅⋅⋅⋅ D′. Oskil\ky D — metacykliçna hrupa, to D′ — cykliçna hrupa i
G′′ = P′ ⋅⋅⋅⋅ D′′ ⋅⋅⋅⋅ [ P, D′ ] = P ⋅⋅⋅⋅ [ P, D′ ] ≤ P. Takym çynom, G′′′ ≤ P′, G(
iv
) ≤ P′′ = 1.
Perßu çastynu teoremy dovedeno.
Nexaj G — nil\potentna hrupa. Za vidomymy rezul\tatamy [10, s. 400] vona
ma[ periodyçnu çastynu T ( G ). MoΩlyvi vypadky:
1) T ( G ) = 1;
2) T ( G ) ≠ 1.
Vypadok 1. Nexaj G ′ ≠ 1. V nil\potentnij hrupi G bez skrutu znajdut\sq
taki elementy a ta b, wo [ a, b ] = c ∈ Z ( G ), | c | = ∞. Poklademo H = 〈 a, b 〉. Za
tverdΩennqm 1.2.1 [11] dlq dovil\noho natural\noho n [ an, b ] = [ a, b ]
n = cn ≠ 1.
Tomu H = ( 〈 c 〉 × 〈 a 〉 ) l 〈 b 〉. Zrozumilo, wo H > N, de N = ( 〈 cp
2
〉 × 〈 ap
〉 ) l 〈 bp
〉
— nemetacykliçna hrupa. Za lemog 2 [1] N normal\na v G i G / M — dedekin-
dova hrupa, komutant qko] mistyt\ sumiΩnyj klas N ⋅⋅⋅⋅ c porqdku p2
, wo supere-
çyt\ teoremi 12.5.4 [12]. Takym çynom, u vypadku 1 G′ = 1 i teoremu dovedeno.
Vypadok 2. U c\omu vypadku za lemog 1 [1] i rozhlqnutym vypadkom 1 ma[mo
G′ ≤ T ( G ). Prypustymo, wo G mistyt\ taki nemetacykliçni pidhrupy A ta B,
dlq qkyx A ∩ B = 1. Todi za lemog 2 [1] A normal\na v G, B normal\na v G,
G / A ta G / B — dedekindovi hrupy, a otΩe, | ( G / A ) ′ | ≤ 2 i | ( G / B ) ′ | ≤ 2.
Nexaj G* = G / A × G / B. Oskil\ky A ∩ B = 1, to za teoremog Remaka [9,
s. 54] G vklada[t\sq v G*
. Tomu | G′ | ≤ 4, i v c\omu vypadku teoremu dovedeno.
Nexaj G ne mistyt\ zhadanyx pidhrup A ta B. Todi za poperednim v T ( G )
dovil\na cilkom faktoryzovana abeleva pidhrupa [ skinçennog hrupog, wo ne
mistyt\ pidhrup porqdku p q r, de p, q, r — neobov’qzkovo rizni prosti çysla.
Za teoremog 1.2 [6] i teoremog 1.9 [6] T ( G ) — çernikovs\ka hrupa z povnog ças-
tynog R, R ≤ Z ( G ). Za prypuwennqm R ne ma[ dvox riznyx kvazicykliçnyx
pidhrup. Pry | T ( G ) | < ∞ teorema [ oçevydnog. Nexaj | T ( G ) | = ∞. Todi za
poperednim povna çastyna R hrupy T ( G ) [ kvazicykliçnog p-hrupog. Pry
T ( G ) = G R ≤ Z ( G ) i G — skinçenna nad centrom hrupa, u qko], qk vidomo,
| G′ | ≤ ∞. Tomu v podal\ßomu ma[mo T ( G ) < G, R ≤ G′ i R ≤ Z ( G ).
Nexaj D — pidhrupa, porodΩena deqkym ßarom elementiv iz T ( G ), wo mis-
tyt\ po odnomu iz predstavnykiv sumiΩnyx klasiv T ( G ) / R. Todi T ( G ) = R ⋅⋅⋅⋅ D,
de D — xarakterystyçna pidhrupa iz T ( G ) i | D | < ∞. Qkwo G / D — abeleva
hrupa, to R ne naleΩyt\ G′, wo supereçyt\ vyboru. OtΩe, G / D ne moΩe bu-
ty navit\ dedekindovog hrupog, a tomu G / D ne moΩe buty rozßyrennqm svo[]
central\no] kvazicykliçno] pidhrupy T ( G ) / D za dopomohog lokal\no cykliç-
no] hrupy bez skrutu ( G / D ) / ( T ( G ) / D ). Zrozumilo, wo sylovs\ka p-pidhrupa iz
G [ abelevog hrupog. Z c\oho vyplyva[, wo v G znajdut\sq taki elementy a
ta b, dlq qkyx | a | ∈ { pα, ∞ }, | b | ∈ { pβ, ∞ }, α > 0, β > 0, | a | = ∞, abo | b | =
= ∞, [ a, b ] = c, c ∈ R, | c | = p3
.
Poklademo H = 〈 a, b 〉. Todi bez porußennq zahal\nosti H = ( 〈 c 〉 × 〈 a 〉 ) l
l 〈 b 〉, H > N , de N = ( 〈 cp
2
〉 × 〈 a 〉 ) l 〈 bp
2
〉 — nemetacykliçna hrupa, | b | = ∞ .
OtΩe, qk i raniße, oderΩaly supereçnist\, wo G / N — dedekindova hrupa. Cq
supereçnist\ zaverßu[ dovedennq teoremy. Vsi vypadky rozhlqnuto.
Teoremu dovedeno.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
LOKAL|NO STUPINÇASTI HRUPY Z NORMAL|NYMY … 135
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| id | umjimathkievua-article-3296 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:39:50Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-32962020-03-18T19:50:22Z Locally graded groups with normal nonmetacyclic subgroups Локально ступінчасті групи з нормальними неметациклічними підгрупами Kovalenko, V. I. Коваленко, В. І. We establish the solvability of locally graded groups with normal nonmetacyclic subgroups and prove that the degree of solvability does not exceed 4. Установлена разрешимость локально ступенчатых групп с нормальными неметациклическими подгруппами и отмечено, что ступень разрешимости не превышает числа 4. Institute of Mathematics, NAS of Ukraine 2007-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3296 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 1 (2007); 133–135 Український математичний журнал; Том 59 № 1 (2007); 133–135 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3296/3337 https://umj.imath.kiev.ua/index.php/umj/article/view/3296/3338 Copyright (c) 2007 Kovalenko V. I. |
| spellingShingle | Kovalenko, V. I. Коваленко, В. І. Locally graded groups with normal nonmetacyclic subgroups |
| title | Locally graded groups with normal nonmetacyclic subgroups |
| title_alt | Локально ступінчасті групи з нормальними неметациклічними підгрупами |
| title_full | Locally graded groups with normal nonmetacyclic subgroups |
| title_fullStr | Locally graded groups with normal nonmetacyclic subgroups |
| title_full_unstemmed | Locally graded groups with normal nonmetacyclic subgroups |
| title_short | Locally graded groups with normal nonmetacyclic subgroups |
| title_sort | locally graded groups with normal nonmetacyclic subgroups |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3296 |
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