On permuteral subgroups in finite groups
The permutizer of a subgroup H in a group G is defined as the subgroup generated by all cyclic subgroups of G that permute with H . Call H permuteral in G if the permutizer of H in G coincides with G ; H is called strongly permuteral in G if the permutizer of H in U coincides with U for every subgro...
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| Published in | Siberian mathematical journal Vol. 55; no. 2; pp. 230 - 238 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Moscow
Pleiades Publishing
01.03.2014
|
| Subjects | |
| Online Access | Get full text |
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| Abstract | The permutizer of a subgroup
H
in a group
G
is defined as the subgroup generated by all cyclic subgroups of
G
that permute with
H
. Call
H
permuteral in
G
if the permutizer of
H
in
G
coincides with
G
;
H
is called strongly permuteral in
G
if the permutizer of
H
in
U
coincides with
U
for every subgroup
U
of
G
containing
H
. We study the finite groups with given systems of permuteral and strongly permuteral subgroups and find some new characterizations of w-supersoluble and supersoluble groups. |
|---|---|
| AbstractList | The permutizer of a subgroup
H
in a group
G
is defined as the subgroup generated by all cyclic subgroups of
G
that permute with
H
. Call
H
permuteral in
G
if the permutizer of
H
in
G
coincides with
G
;
H
is called strongly permuteral in
G
if the permutizer of
H
in
U
coincides with
U
for every subgroup
U
of
G
containing
H
. We study the finite groups with given systems of permuteral and strongly permuteral subgroups and find some new characterizations of w-supersoluble and supersoluble groups. |
| Author | Vasil’ev, A. F. Vasil’ev, V. A. Vasil’eva, T. I. |
| Author_xml | – sequence: 1 givenname: A. F. surname: Vasil’ev fullname: Vasil’ev, A. F. email: formation56@mail.ru organization: Francisk Skorina Gomel State University – sequence: 2 givenname: V. A. surname: Vasil’ev fullname: Vasil’ev, V. A. organization: Francisk Skorina Gomel State University – sequence: 3 givenname: T. I. surname: Vasil’eva fullname: Vasil’eva, T. I. organization: Belarusian State University of Transport |
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| Cites_doi | 10.1017/S000497270004274X 10.1007/BF01650529 10.1006/jabr.1996.6932 10.1134/S0037446612010041 10.1007/978-3-662-21943-0 10.1007/s11202-010-0099-z 10.1007/BF01446920 10.1515/9783110870138 10.1081/AGB-120016008 10.1016/j.jpaa.2008.03.004 10.1081/AGB-200047439 |
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| Keywords | w-supersoluble group supersoluble group finite group permuteral subgroup permutizer of a subgroup ℙ-subnormal subgroup |
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| References | Ballester-BolinchesAEsteban-RomeroROn a question of Beidleman and RobinsonComm. Algebra200230125757577010.1081/AGB-1200160081018.200131941921 HeinekenHProducts of finite nilpotent groupsMath. Ann.1990287164365210.1007/BF014469200687.200131066821 CoxeterH S MMoserW O JGenerators and Relations for Discrete Groups [Russian translation]1980MoscowNauka10.1007/978-3-662-21943-0 DoerkKHawkesTFinite Soluble Groups1992Berlin and New YorkWalter de Gruyter10.1515/97831108701380753.20001 ZhangJA note on finite groups satisfying the permutizer conditionSci. Bull.1986313633650603.20016 BeidlemanJ CRobinsonD J SOn finite groups satisfying the permutizer conditionJ. Algebra1997191268670310.1006/jabr.1996.69320880.200151448814 ShemetkovL AFormations of Finite Groups [in Russian]1978MoscowNauka KniahinaVMonakhovVOn supersolvability of finite groups with P-subnormal subgroupsInt. J. Group Theory20132421293065875 LiuXWangYaImplications of permutizers of some subgroups in finite groupsComm. Algebra20053355956510.1081/AGB-2000474391075.200102124345 QiaoShQianGWangYaInfluence of permutizers of subgroups on the structure of finite groupsJ. Pure Appl. Algebra2008212102307231310.1016/j.jpaa.2008.03.0041156.200142426510 PenningtonE ATrifactorisable groupsBull. Austral. Math. Soc.19738346146910.1017/S000497270004274X0249.20013318307 WielandtHÜber Produkte von nilpotenten Gruppen. IIIJ. Math.195824B6116180084.02904136656 Vasil’evA FVasil’evaT ITyutyanovV NOn the finite groups of supersoluble typeSiberian Math. J.20105161004101210.1007/s11202-010-0099-z1226.200132797596 KegelO HProdukte nilpotenter GruppenArch. Math. (Basel)1961121909310.1007/BF016505290099.01401133365 WeinsteinMBetween Nilpotent and Solvable1982PassaicPolygonal Publ. House0488.20001 Vasil’evA FVasil’evaT ITyutyanovV NOn the products of P-subnormal subgroups of finite groupsSiberian Math. J.2012531475410.1134/S00374466120100411242.200272962189 K Doerk (2005_CR8) 1992 H Heineken (2005_CR11) 1990; 287 E A Pennington (2005_CR12) 1973; 8 V Kniahina (2005_CR16) 2013; 2 A Ballester-Bolinches (2005_CR4) 2002; 30 Sh Qiao (2005_CR6) 2008; 212 J Zhang (2005_CR2) 1986; 31 A F Vasil’ev (2005_CR9) 2010; 51 H Wielandt (2005_CR14) 1958; 2 A F Vasil’ev (2005_CR10) 2012; 53 H S M Coxeter (2005_CR15) 1980 (2005_CR1) 1982 O H Kegel (2005_CR13) 1961; 12 J C Beidleman (2005_CR3) 1997; 191 X Liu (2005_CR5) 2005; 33 L A Shemetkov (2005_CR7) 1978 |
| References_xml | – volume: 8 start-page: 461 issue: 3 year: 1973 ident: 2005_CR12 publication-title: Bull. Austral. Math. Soc. doi: 10.1017/S000497270004274X contributor: fullname: E A Pennington – volume: 2 start-page: 21 issue: 4 year: 2013 ident: 2005_CR16 publication-title: Int. J. Group Theory contributor: fullname: V Kniahina – volume-title: Formations of Finite Groups [in Russian] year: 1978 ident: 2005_CR7 contributor: fullname: L A Shemetkov – volume: 12 start-page: 90 issue: 1 year: 1961 ident: 2005_CR13 publication-title: Arch. Math. (Basel) doi: 10.1007/BF01650529 contributor: fullname: O H Kegel – volume: 191 start-page: 686 issue: 2 year: 1997 ident: 2005_CR3 publication-title: J. Algebra doi: 10.1006/jabr.1996.6932 contributor: fullname: J C Beidleman – volume: 53 start-page: 47 issue: 1 year: 2012 ident: 2005_CR10 publication-title: Siberian Math. J. doi: 10.1134/S0037446612010041 contributor: fullname: A F Vasil’ev – volume: 2 start-page: 611 issue: 4B year: 1958 ident: 2005_CR14 publication-title: J. Math. contributor: fullname: H Wielandt – volume-title: Generators and Relations for Discrete Groups [Russian translation] year: 1980 ident: 2005_CR15 doi: 10.1007/978-3-662-21943-0 contributor: fullname: H S M Coxeter – volume: 51 start-page: 1004 issue: 6 year: 2010 ident: 2005_CR9 publication-title: Siberian Math. J. doi: 10.1007/s11202-010-0099-z contributor: fullname: A F Vasil’ev – volume: 287 start-page: 643 issue: 1 year: 1990 ident: 2005_CR11 publication-title: Math. Ann. doi: 10.1007/BF01446920 contributor: fullname: H Heineken – volume-title: Finite Soluble Groups year: 1992 ident: 2005_CR8 doi: 10.1515/9783110870138 contributor: fullname: K Doerk – volume-title: Between Nilpotent and Solvable year: 1982 ident: 2005_CR1 – volume: 30 start-page: 5757 issue: 12 year: 2002 ident: 2005_CR4 publication-title: Comm. Algebra doi: 10.1081/AGB-120016008 contributor: fullname: A Ballester-Bolinches – volume: 212 start-page: 2307 issue: 10 year: 2008 ident: 2005_CR6 publication-title: J. Pure Appl. Algebra doi: 10.1016/j.jpaa.2008.03.004 contributor: fullname: Sh Qiao – volume: 33 start-page: 559 year: 2005 ident: 2005_CR5 publication-title: Comm. Algebra doi: 10.1081/AGB-200047439 contributor: fullname: X Liu – volume: 31 start-page: 363 year: 1986 ident: 2005_CR2 publication-title: Sci. Bull. contributor: fullname: J Zhang |
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| Snippet | The permutizer of a subgroup
H
in a group
G
is defined as the subgroup generated by all cyclic subgroups of
G
that permute with
H
. Call
H
permuteral in
G
if... |
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| Title | On permuteral subgroups in finite groups |
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