Movatterモバイル変換

[0]ホーム
URL:

×

zbMATH Open — the first resource for mathematics

from until
Reset all

Examples

GeometrySearch for the termGeometry inany field. Queries arecase-independent.
Funct*Wildcard queries are specified by* (e .g.functions,functorial, etc.). Otherwise the search isexact.''Topological group'':Phrases (multi - words) should be set in''straight quotation marks''.
au: Bourbaki & ti: AlgebraSearch forauthorBourbaki andtitleAlgebra. Theand-operator & is default and can be omitted.
Chebyshev | TschebyscheffTheor-operator| allows to search forChebyshev orTschebyscheff.
Quasi* map* py: 1989The resulting documents havepublicationyear1989.
so:Eur* J* Mat* Soc* cc:14Search for publications in a particularsource with aMathematics SubjectClassificationcode in14.
cc:*35 ! any:ellipticSearch for documents about PDEs (prefix with * to search only primary MSC); the not-operator ! eliminates all results containing the wordelliptic.
dt: b & au: HilbertThedocumenttype is set tobooks; alternatively:j forjournal articles,a forbookarticles.
py: 2000 - 2015 cc:(94A | 11T)Numberranges when searching forpublicationyear are accepted . Terms can be grouped within( parentheses).
la: chineseFind documents in a givenlanguage .ISO 639 - 1 (opens in new tab) language codes can also be used.
st: c r sFind documents that arecited, havereferences and are from asingle author.

Fields

ab Text from the summary or review (for phrases use “. ..”)
an zbMATH ID, i.e.: preliminary ID, Zbl number, JFM number, ERAM number
any Includes ab, au, cc, en, rv, so, ti, ut
arxiv arXiv preprint number
au Name(s) of the contributor(s)
br Name of a person with biographic references (to find documents about the life or work)
cc Code from the Mathematics Subject Classification (prefix with* to search only primary MSC)
ci zbMATH ID of a document cited in summary or review
db Database: documents in Zentralblatt für Mathematik/zbMATH Open (db:Zbl), Jahrbuch über die Fortschritte der Mathematik (db:JFM), Crelle's Journal (db:eram), arXiv (db:arxiv)
dt Type of the document: journal article (dt:j), collection article (dt:a), book (dt:b)
doi Digital Object Identifier (DOI)
ed Name of the editor of a book or special issue
en External document ID: DOI, arXiv ID, ISBN, and others
in zbMATH ID of the corresponding issue
la Language (use name, e.g.,la:French, orISO 639-1, e.g.,la:FR)
li External link (URL)
na Number of authors of the document in question. Interval search with “-”
pt Reviewing state: Reviewed (pt:r), Title Only (pt:t), Pending (pt:p), Scanned Review (pt:s)
pu Name of the publisher
py Year of publication. Interval search with “-”
rft Text from the references of a document (for phrases use “...”)
rn Reviewer ID
rv Name or ID of the reviewer
se Serial ID
si swMATH ID of software referred to in a document
so Bibliographical source, e.g., serial title, volume/issue number, page range, year of publication, ISBN, etc.
st State: is cited (st:c), has references (st:r), has single author (st:s)
sw Name of software referred to in a document
ti Title of the document
ut Keywords

Operators

a & bLogical and (default)
a | bLogical or
!abLogical not
abc*Right wildcard
ab cPhrase
(ab c)Term grouping

See also ourGeneral Help.

Overgroups of cyclic Sylow subgroups of linear groups.(English)Zbl 1162.20033

R. Guralnick,T. Penttila,C. E. Praeger andJ. Saxl [Proc. Lond. Math. Soc., III. Ser. 78, No. 1, 167-214 (1999;Zbl 1041.20035)] classified the subgroups of finite linear groups which intersect nontrivially with a cyclic Sylow subgroup. This classification has been applied in a number of situations.
The main purpose of this paper is to explore possible applications of this result in geometry; as the authors put it, “to produce tools purpose-built for easier application”. The main results are to be applied in two sequel articles to provide information concerningeggs of projective spaces, andovoids,spreads and\(m\)-systems of polar spaces, [seeJ. Bamberg andT. Penttila, Innov. Incidence Geom. 4, 1-12 (2006;Zbl 1127.51005)].
The main results can all be viewed as specializations of the result of Guralnick et al.: we are given a subgroup \(G\) of some finite general linear group \(\text{GL}_d(q)\), and some extra numerical information about \(G\) (eg., we are told some information about the order of \(G\), or that \(G\) contains a subgroup of a given index). Then the different possible subgroups \(G\) satisfying the hypotheses are listed. The lists break up into several families (eg., classical examples, reducible examples, sporadic group examples), with the possibilities for each case listed, along with complete information about the numerical data associated with each possibility (eg., which values of \(d\) and \(q\) can arise).
The final section of the paper applies the results to prove a conjecture of Cameron and Liebler concerning collineation groups of finite projective spaces.

MSC:

20G40 Linear algebraic groups over finite fields
20C33 Representations of finite groups of Lie type
20E28 Maximal subgroups
51E20 Combinatorial structures in finite projective spaces
20G05 Representation theory for linear algebraic groups
20D06 Simple groups: alternating groups and groups of Lie type
20C34 Representations of sporadic groups

Software:

GAP

Cite

References:

[1]Abhyankar S. S., Proc. Amer. Math. Soc. 124 (10) pp 2967– (1996) ·Zbl 0866.12004 ·doi:10.1090/S0002-9939-96-03471-5
[2]Baddeley R. W., J. Algebra 263 (2) pp 294– (2003) ·Zbl 1024.20002 ·doi:10.1016/S0021-8693(03)00113-3
[3]Bamberg J., Innov. Incidence Geom. 4 pp 1– (2006)
[4]Bereczky Á., J. Algebra 234 (1) pp 187– (2000) ·Zbl 0971.20024 ·doi:10.1006/jabr.2000.8458
[5]Cameron P. J., J. Algebra 60 (2) pp 384– (1979) ·Zbl 0417.20044 ·doi:10.1016/0021-8693(79)90090-5
[6]Cameron P. J., Linear Algebra Appl. 46 pp 91– (1982) ·Zbl 0504.05015 ·doi:10.1016/0024-3795(82)90029-5
[7]Conway J. H., Atlas of Finite Groups (1985)
[8]Cooperstein B. N., Israel J. Math. 30 (3) pp 213– (1978) ·Zbl 0383.20027 ·doi:10.1007/BF02761072
[9]Cooperstein B. N., J. Algebra 70 (1) pp 23– (1981) ·Zbl 0459.20007 ·doi:10.1016/0021-8693(81)90241-6
[10]Dixon J. D., Permutation Groups (1996) ·Zbl 0951.20001 ·doi:10.1007/978-1-4612-0731-3
[11]The GAP Group . ( 2005 ). GAP – Groups, Algorithms, and Programming, version 4.4.10. 2007.http://www.gap-system.org .
[12]Guralnick R., Proc. London Math. Soc. (3) 78 (1) pp 167– (1999) ·Zbl 1041.20035 ·doi:10.1112/S0024611599001616
[13]Guralnick R. M., J. Algebra 234 (2) pp Wielandt– (2000)
[14]Hering C., Geometriae Dedicata 2 pp 425– (1974) ·Zbl 0292.20045 ·doi:10.1007/BF00147570
[15]Hiss G., LMS J. Comput. Math. 4 pp 22– (2001) ·Zbl 0979.20012 ·doi:10.1112/S1461157000000796
[16]Jansen C., An Atlas of Brauer Characters (1995) ·Zbl 0831.20001
[17]Kantor W. M., Israel J. Math. 14 pp 229– (1973) ·Zbl 0262.20055 ·doi:10.1007/BF02764881
[18]Kleidman P. B., J. Algebra 110 (1) pp 173– (1987) ·Zbl 0623.20031 ·doi:10.1016/0021-8693(87)90042-1
[19]Kleidman , P. B. ( 1987b ). The Subgroup Structure of Some Finite Simple Groups . PhD Thesis , University of Cambridge . ·Zbl 0595.20027
[20]Kleidman P. B., J. Algebra 117 (1) pp 117– (1988) ·Zbl 0652.51013 ·doi:10.1016/0021-8693(88)90244-X
[21]Kleidman P. B., J. Algebra 117 (1) pp 30– (1988) ·Zbl 0651.20020 ·doi:10.1016/0021-8693(88)90239-6
[22]Kleidman P., The Subgroup Structure of the Finite Classical Groups (1990) ·Zbl 0697.20004 ·doi:10.1017/CBO9780511629235
[23]O’Brien E. A., Finite Geometries, Groups and Computation pp 163– (2006)
[24]Penttila , T. ( 1985 ). Collineations and Configurations in Projective Spaces . D. Phil. thesis , University of Oxford .
[25]Suzuki M., Ann. Math. 75 (2) pp 105– (1962) ·Zbl 0106.24702 ·doi:10.2307/1970423
[26]Valentini R. C. Madan, J. Reine Angew. Math. 318 pp 156– (1980)
[27]Zsigmondy K., Monatsh. für Math. u. Phys. 3 pp 265– (1892) ·JFM 24.0176.02 ·doi:10.1007/BF01692444
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
© 2025FIZ Karlsruhe GmbHPrivacy PolicyLegal NoticesTerms & Conditions
  • Mastodon logo
 (opens in new tab)
[8]ページ先頭
©2009-2025 Movatter.jp