On finite generalized quadrangles with PSL(2,q) as an automorphism group
Let S be a finite thick generalized quadrangle, and suppose that G is an automorphism group of S . If G acts primitively on both the points and lines of S , then it is known that G must be almost simple. In this paper, we show that if the socle of G is PSL ( 2 , q ) with q ≥ 4 , then q = 9 and S is...
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| Published in: | Designs, codes, and cryptography Vol. 91; no. 6; pp. 2347 - 2364 |
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| Language: | English |
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01.06.2023
Springer Nature B.V |
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| Abstract | Let
S
be a finite thick generalized quadrangle, and suppose that
G
is an automorphism group of
S
. If
G
acts primitively on both the points and lines of
S
, then it is known that
G
must be almost simple. In this paper, we show that if the socle of
G
is
PSL
(
2
,
q
)
with
q
≥
4
, then
q
=
9
and
S
is the unique generalized quadrangle of order 2. |
|---|---|
| AbstractList | Let
S
be a finite thick generalized quadrangle, and suppose that
G
is an automorphism group of
S
. If
G
acts primitively on both the points and lines of
S
, then it is known that
G
must be almost simple. In this paper, we show that if the socle of
G
is
PSL
(
2
,
q
)
with
q
≥
4
, then
q
=
9
and
S
is the unique generalized quadrangle of order 2. Let S be a finite thick generalized quadrangle, and suppose that G is an automorphism group of S. If G acts primitively on both the points and lines of S, then it is known that G must be almost simple. In this paper, we show that if the socle of G is PSL(2,q) with q≥4, then q=9 and S is the unique generalized quadrangle of order 2. |
| Author | Lu, Jianbing Feng, Tao |
| Author_xml | – sequence: 1 givenname: Tao surname: Feng fullname: Feng, Tao organization: School of Mathematical Sciences, Zhejiang University – sequence: 2 givenname: Jianbing orcidid: 0000-0002-2699-4256 surname: Lu fullname: Lu, Jianbing email: jianbinglu@zju.edu.cn organization: School of Mathematical Sciences, Zhejiang University |
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| Cites_doi | 10.1007/BF01263523 10.4171/066 10.1016/j.jcta.2016.11.008 10.1016/j.disc.2021.112291 10.1016/j.jcta.2008.02.004 10.1017/nmj.2017.35 10.37236/5510 10.1007/978-3-642-64981-3 10.1017/CBO9781139192576 10.1017/CBO9781139059060 10.1007/BF00182417 10.1016/j.jcta.2012.04.005 10.1016/0021-8693(64)90028-6 10.1112/plms/s3-63.2.266 10.1002/jcd.21411 10.1515/jgth-2016-0042 10.1007/BF02684706 10.37236/1076 10.1007/978-3-0348-8827-1 |
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| References | CR19 CR16 CR15 Burness, Giudici (CR10) 2016 Bamberg, Glasby, Popiel, Praeger, Schneider (CR4) 2017; 147 Huppert (CR17) 1967 Bamberg, Evans (CR1) 2021; 344 Ghinelli (CR14) 1992; 41 Isaacs (CR18) 1976 Cameron, Omidi, Tayeh-Rezaie (CR11) 2006; 13 Schneider, Van Maldeghem (CR24) 2008; 115 Dickson (CR12) 1958 Payne, Thas (CR23) 2009 Van Maldeghem (CR21) 1998 Morgan, Popiel (CR22) 2016; 23 Tits (CR25) 1959; 2 CR7 Feit, Higman (CR13) 1964; 1 Bray, Holt, Roney-Dougal (CR8) 2013 Liebeck, Saxl (CR20) 1991; 63 Bamberg, Popiel, Praeger (CR6) 2017; 234 Buekenhout, Van Maldeghem (CR9) 1994; 52 Bamberg, Giudici, Morris, Royle, Spiga (CR2) 2012; 119 Bamberg, Glasby, Popiel, Praeger (CR3) 2016; 24 Bamberg, Popiel, Praeger (CR5) 2017; 20 |
| References_xml | – volume: 52 start-page: 41 issue: 1 year: 1994 end-page: 51 ident: CR9 article-title: Finite distance-transitive generalized polygons publication-title: Geom. Dedicata doi: 10.1007/BF01263523 – year: 2009 ident: CR23 publication-title: Finite Generalized Quadrangles doi: 10.4171/066 – volume: 147 start-page: 186 year: 2017 end-page: 204 ident: CR4 article-title: Point-primitive generalised hexagons and octagons publication-title: J. Combin. Theory Ser. A doi: 10.1016/j.jcta.2016.11.008 – year: 1958 ident: CR12 publication-title: Linear Groups: With an Exposition of the Galois Field Theory – ident: CR16 – volume: 344 start-page: 112291 year: 2021 ident: CR1 article-title: No sporadic almost simple group acts primitively on the points of a generalised quadrangle publication-title: Discret. Math. doi: 10.1016/j.disc.2021.112291 – volume: 115 start-page: 1436 issue: 8 year: 2008 end-page: 1455 ident: CR24 article-title: Primitive flag-transitive generalized hexagons and octagons publication-title: J. Combin. Theory Ser. A doi: 10.1016/j.jcta.2008.02.004 – volume: 234 start-page: 87 year: 2017 ident: CR6 article-title: Simple groups, product actions, and generalized quadrangles publication-title: Nagoya Math. J. doi: 10.1017/nmj.2017.35 – volume: 23 start-page: 1 year: 2016 end-page: 34 ident: CR22 article-title: Generalised polygons admitting a point-primitive almost simple group of Suzuki or Ree type publication-title: Electron. J. Combin. doi: 10.37236/5510 – year: 1967 ident: CR17 publication-title: Endliche Gruppen I doi: 10.1007/978-3-642-64981-3 – year: 2013 ident: CR8 publication-title: The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, London Mathematical Society Lecture Note Series 407 doi: 10.1017/CBO9781139192576 – year: 2016 ident: CR10 publication-title: Classical Groups, Derangements and Primes, Australian Mathematical Society Lecture Series doi: 10.1017/CBO9781139059060 – volume: 41 start-page: 165 year: 1992 end-page: 174 ident: CR14 article-title: Regular groups on generalized quadrangles and nonabelian difference sets with multiplier publication-title: Geom. Dedicata doi: 10.1007/BF00182417 – ident: CR19 – volume: 119 start-page: 1479 issue: 7 year: 2012 end-page: 1499 ident: CR2 article-title: Generalised quadrangles with a group of automorphisms acting primitively on points and lines publication-title: J. Combin. Theory Ser. A doi: 10.1016/j.jcta.2012.04.005 – ident: CR15 – volume: 1 start-page: 114 year: 1964 end-page: 131 ident: CR13 article-title: The nonexistence of certain generalized polygons publication-title: J. Algebra doi: 10.1016/0021-8693(64)90028-6 – volume: 63 start-page: 266 year: 1991 end-page: 314 ident: CR20 article-title: Minimal degrees of primitive permutation groups, with an application to Monodromy groups of covers of Riemann surfaces publication-title: Proc. Lond. Math. Soc. (3) doi: 10.1112/plms/s3-63.2.266 – volume: 24 start-page: 151 year: 2016 end-page: 164 ident: CR3 article-title: Generalized quadrangles and transitive pseudo-hyperovals publication-title: J. Combin. Des. doi: 10.1002/jcd.21411 – ident: CR7 – volume: 20 start-page: 269 issue: 2 year: 2017 end-page: 287 ident: CR5 article-title: Point-primitive, line-transitive generalised quadrangles of holomorph type publication-title: J. Group Theory doi: 10.1515/jgth-2016-0042 – volume: 2 start-page: 13 year: 1959 end-page: 60 ident: CR25 article-title: Sur la trialité et certains groupes qui s’en déduisent publication-title: Publ. Math. IHÉS doi: 10.1007/BF02684706 – year: 1976 ident: CR18 publication-title: Character Theory of Finite Groups – volume: 13 start-page: R50 year: 2006 ident: CR11 article-title: -Designs from publication-title: Electron. J. Combin. doi: 10.37236/1076 – year: 1998 ident: CR21 publication-title: Generalized Polygons doi: 10.1007/978-3-0348-8827-1 |
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| Snippet | Let
S
be a finite thick generalized quadrangle, and suppose that
G
is an automorphism group of
S
. If
G
acts primitively on both the points and lines of
S
,... Let S be a finite thick generalized quadrangle, and suppose that G is an automorphism group of S. If G acts primitively on both the points and lines of S, then... |
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| SubjectTerms | Automorphisms Coding and Information Theory Computer Science Cryptology Discrete Mathematics in Computer Science Geometry Inequality Polygons |
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| Title | On finite generalized quadrangles with PSL(2,q) as an automorphism group |
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