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SymmetricFinite Generalised Polygons
John Bamberg
Centre for the Mathematics of Symmetry and Computation,The University of Western Australia
John Bamberg Symmetric Finite Generalised Polygons
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Theorem (B. Kerekjarto (1941))
Every triply transitive group of continuous transformations of thecircle or the sphere is permutationally isomorphic to
PGL(2,R) (circle)
PGL(2,C) or PGL(2,C) o (complex conj.) (sphere).
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Theorem (T. G. Ostrom and A. Wagner (1959))
A finite projective plane admitting a doubly transitive group ofautomorphisms is Desarguesian.
Quote: W. M. Kantor (1993)1
“This was the first time 2-transitivity produced a complete classification offinite geometries. Since then the notion of a geometric classification in terms ofa group-theoretic hypothesis has become commonplace. That was not the case35 years ago, and it is a measure of these papers’ influence that this type ofhypothesis is now regarded as a natural extension of Klein’s Erlangen program.”
1‘2-transitive and flag-transitive designs’, Coding theory, design theory,group theory (Burlington, VT), 13–30, Wiley.
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Theorem (T. G. Ostrom and A. Wagner (1959))
A finite projective plane admitting a doubly transitive group ofautomorphisms is Desarguesian.
Quote: W. M. Kantor (1993)1
“This was the first time 2-transitivity produced a complete classification offinite geometries. Since then the notion of a geometric classification in terms ofa group-theoretic hypothesis has become commonplace. That was not the case35 years ago, and it is a measure of these papers’ influence that this type ofhypothesis is now regarded as a natural extension of Klein’s Erlangen program.”
1‘2-transitive and flag-transitive designs’, Coding theory, design theory,group theory (Burlington, VT), 13–30, Wiley.
John Bamberg Symmetric Finite Generalised Polygons
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Theorem (R. Moufang (1932/33); G. Pickert (1955))
Let Γ be a projective plane and let G 6 Aut(Γ). If for every line `,G(`) acts transitively on the points of Γ \ `,then Γ can be coordinatised by an alternative division ring.
point-wise stabiliser of `
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Theorem (R. Moufang (1932/33); G. Pickert (1955))
Let Γ be a projective plane and let G 6 Aut(Γ). If for every line `,G(`) acts transitively on the points of Γ \ `,then Γ can be coordinatised by an alternative division ring.
point-wise stabiliser of `
John Bamberg Symmetric Finite Generalised Polygons
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Flag
D. G. Higman & J. E. McLaughlin (1961)
Let Γ be a linear space and G 6 Aut(Γ).
G transitive on flags =⇒ G primitive on points.
Primitive
G 6 Sym(Ω) does not preserve a partition of Ω, except the trivial ones:
Ωω : ω ∈ Ω
2-transitive =⇒ 2-homogeneous =⇒ primitive =⇒ quasiprimitive
=⇒ innately transitive =⇒ semiprimitive =⇒ transitive
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W. M. Kantor (1987)
A projective plane π of order x admitting a point-primitiveautomorphism group G is Desarguesian and G > PSL(3, x), or elseG is boring2.
K. Thas and Zagier 2008
A non-Desarguesian projective plane π with Aut(π) point-primitivehas at least 4× 1022 points.
2The number of points (x2 + x + 1) is a prime and G is a regular orFrobenius group of order dividing (x2 + x + 1)(x + 1) or (x2 + x + 1)x .
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B. Xia (2018)
If there is a finite non-Desarguesian flag-transitive projective planeof order x with v = x2 + x + 1 points, then
v is prime with m ≡ 8 (mod 24), and
there exists a finite field F of characteristic 3, and melements, satisfying certain polynomial equations.
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N. Gill (2016)
If G acts transitively on a finite non-Desarguesian projective plane,then
the Sylow 2-subgroups of G are cyclic or generalisedquaternion, and
if G is insoluble, then G/O(G ) ∼= SL2(5),SL2(5).2.
Conjecture; D. Hughes (1959)
A finite projective plane with a transitive automorphism group isDesarguesian.
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N. Gill (2016)
If G acts transitively on a finite non-Desarguesian projective plane,then
the Sylow 2-subgroups of G are cyclic or generalisedquaternion, and
if G is insoluble, then G/O(G ) ∼= SL2(5),SL2(5).2.
Conjecture; D. Hughes (1959)
A finite projective plane with a transitive automorphism group isDesarguesian.
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J. Tits, 1959
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Generalised polygons
1
3 5
2 4
6
7
1
2
5
6
7
4
3
123
257
356
167
347
145
246
Generalised n-gon:Incidence graph has girth = 2× diameter = 2n.
Feit-Higman Theorem (1964):Thick =⇒ n ∈ 2, 3, 4, 6, 8.
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Generalised polygons
1
3 5
2 4
6
7
1
2
5
6
7
4
3
123
257
356
167
347
145
246
Generalised n-gon:Incidence graph has girth = 2× diameter = 2n.
Feit-Higman Theorem (1964):Thick =⇒ n ∈ 2, 3, 4, 6, 8.
John Bamberg Symmetric Finite Generalised Polygons
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Generalised polygons
Equivalent definition
(i) there are no ordinary k-gons for 2 6 k < n,
(ii) any two elements are contained in some ordinary n-gon.
order (s, t)
every line has s + 1 points,every point lies on t + 1 lines.
thick if s, t > 2.
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Generalised polygons
Equivalent definition
(i) there are no ordinary k-gons for 2 6 k < n,
(ii) any two elements are contained in some ordinary n-gon.
order (s, t)
every line has s + 1 points,every point lies on t + 1 lines.
thick if s, t > 2.
John Bamberg Symmetric Finite Generalised Polygons
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Classical examples
3 projective planes
Desarguesian planes → PSL(3, q).
4 generalised quadrangles
polar spaces associated with PSp(4, q),PSU(4, q) and PSU(5, q), and their duals.
6 generalised hexagons
geometries for G2(q) and 3D4(q).
8 generalised octagons
geometries for 2F4(q).
Many other examples of projective planes and generalisedquadrangles known.
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Classical examples
3 projective planesDesarguesian planes → PSL(3, q).
4 generalised quadranglespolar spaces associated with PSp(4, q),PSU(4, q) and PSU(5, q), and their duals.
6 generalised hexagonsgeometries for G2(q) and 3D4(q).
8 generalised octagonsgeometries for 2F4(q).
Many other examples of projective planes and generalisedquadrangles known.
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Classical examples
3 projective planesDesarguesian planes → PSL(3, q).
4 generalised quadranglespolar spaces associated with PSp(4, q),PSU(4, q) and PSU(5, q), and their duals.
6 generalised hexagonsgeometries for G2(q) and 3D4(q).
8 generalised octagonsgeometries for 2F4(q).
Many other examples of projective planes and generalisedquadrangles known.
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John Bamberg Symmetric Finite Generalised Polygons
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Importance
Building blocks of a building.
Important to groups of Lie type, in many ways.
Missing piece of the classification of spherical buildings.
Many connections to other things in finite geometry andcombinatorics.
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Symmetry
‘Classical’ =⇒Moufang, flag-transitive, point-primitive, and line-primitive.
Moufang for generalised quadrangles
For each path (v0, v1, v2, v3), the group G[1]v0 ∩ G
[1]v1 ∩ G
[1]v2 acts
transitively on Γ(v3) \ v2.
G[1]vi is the kernel of the action of Gvi on Γ(vi ).
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Projective planes
2-transitive
Flag-transitive
Point-primitive
Point-transitive
Moufang
Antiflag-transitive
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Projective planes
2-transitive
Flag-transitive
Point-primitive
Point-transitive
Moufang
Antiflag-transitive
Classical (Ostrom-Wagner 1959)
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Projective planes
2-transitive
Flag-transitive
Point-primitive
Point-transitive
Moufang
Antiflag-transitive
Classical (Ostrom-Wagner 1959)
Classical or small group (Kantor 1987)
Classical or small group (Kantor 1987)
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Projective planes
2-transitive
Flag-transitive
Point-primitive
Point-transitive
Moufang
Antiflag-transitive
Classical (Ostrom-Wagner 1959)
Classical or small group (Kantor 1987)
Classical or small group (Kantor 1987)
Classical or some control over minimalnormal subgroups (Gill 2007,2016)
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Generalised quadrangles
Rank 3 on points
Flag-transitive
Point or line primitive
Point-distance-transitive
Point and line dist.-transitive
12-Moufang
Moufang
Antiflag-transitive
Locally 2-transitive
Buekenhout, Van Maldeghem (1994)
Kantor (1991)
B., Li, Swartz (2018)
B., Li, Swartz (2019+)
Thas,Thas,VM (1991)
Fong, Seitz (1973)
John Bamberg Symmetric Finite Generalised Polygons
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Generalised quadrangles
Rank 3 on points
Flag-transitive
Point or line primitive
Point-distance-transitive
Point and line dist.-transitive
12-Moufang
Moufang
Antiflag-transitive
Locally 2-transitive
Buekenhout, Van Maldeghem (1994)
Kantor (1991)
B., Li, Swartz (2018)
B., Li, Swartz (2019+)
Thas,Thas,VM (1991)
Fong, Seitz (1973)
John Bamberg Symmetric Finite Generalised Polygons
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Generalised quadrangles
Rank 3 on points
Flag-transitive
Point or line primitive
Point-distance-transitive
Point and line dist.-transitive
12-Moufang
Moufang
Antiflag-transitive
Locally 2-transitive
Buekenhout, Van Maldeghem (1994)
Kantor (1991)
B., Li, Swartz (2018)
B., Li, Swartz (2019+)
Thas,Thas,VM (1991)
Fong, Seitz (1973)
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Generalised quadrangles
Rank 3 on points
Flag-transitive
Point or line primitive
Point-distance-transitive
Point and line dist.-transitive
12-Moufang
Moufang
Antiflag-transitive
Locally 2-transitive
Buekenhout, Van Maldeghem (1994)
Kantor (1991)
B., Li, Swartz (2018)
B., Li, Swartz (2019+)
Thas,Thas,VM (1991)
Fong, Seitz (1973)
John Bamberg Symmetric Finite Generalised Polygons
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Generalised quadrangles
Rank 3 on points
Flag-transitive
Point or line primitive
Point-distance-transitive
Point and line dist.-transitive
12-Moufang
Moufang
Antiflag-transitive
Locally 2-transitive
Buekenhout, Van Maldeghem (1994)
Kantor (1991)
B., Li, Swartz (2018)
B., Li, Swartz (2019+)
Thas,Thas,VM (1991)
Fong, Seitz (1973)
John Bamberg Symmetric Finite Generalised Polygons
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Generalised quadrangles
Rank 3 on points
Flag-transitive
Point or line primitive
Point-distance-transitive
Point and line dist.-transitive
12-Moufang
Moufang
Antiflag-transitive
Locally 2-transitive
Buekenhout, Van Maldeghem (1994)
Kantor (1991)
B., Li, Swartz (2018)
B., Li, Swartz (2019+)
Thas,Thas,VM (1991)
Fong, Seitz (1973)
John Bamberg Symmetric Finite Generalised Polygons
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Generalised quadrangles
Rank 3 on points
Flag-transitive
Point or line primitive
Point-distance-transitive
Point and line dist.-transitive
12-Moufang
Moufang
Antiflag-transitive
Locally 2-transitive
Buekenhout, Van Maldeghem (1994)
Kantor (1991)
B., Li, Swartz (2018)
B., Li, Swartz (2019+)
Thas,Thas,VM (1991)
Fong, Seitz (1973)
John Bamberg Symmetric Finite Generalised Polygons
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The generalised quadrangle of order (3, 5)
Derived from Sp(4, 4)-GQ.
Automorphism group: 26 : (3.A6.2).
Point-primitive
Flag-transitive
Line-imprimitive
• GG (G) p G
p p
H(q)
H(2) H(2)D
GQ(3, 5)
Picture courtesy of James Evans.John Bamberg Symmetric Finite Generalised Polygons
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General results
Fong and Seitz (1973)A finite thick generalised polygon satisfying the Moufang conditionis classical or dual classical.
Buekenhout-Van Maldeghem (1994)
A finite thick generalised polygon with a group actingdistance-transitively on points is classical or GQ(3, 5).
Distance-transitive =⇒ point-primitive.
John Bamberg Symmetric Finite Generalised Polygons
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General results
Fong and Seitz (1973)A finite thick generalised polygon satisfying the Moufang conditionis classical or dual classical.
Buekenhout-Van Maldeghem (1994)
A finite thick generalised polygon with a group actingdistance-transitively on points is classical or GQ(3, 5).
Distance-transitive =⇒ point-primitive.
John Bamberg Symmetric Finite Generalised Polygons
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General results
Fong and Seitz (1973)A finite thick generalised polygon satisfying the Moufang conditionis classical or dual classical.
Buekenhout-Van Maldeghem (1994)
A finite thick generalised polygon with a group actingdistance-transitively on points is classical or GQ(3, 5).
Distance-transitive =⇒ point-primitive.
John Bamberg Symmetric Finite Generalised Polygons
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For generalised quadrangles
B., Giudici, Morris, Royle, Spiga (2012)If G acts primitively on the points and lines of a thick GQ then:
G is almost simple3.
If G is also flag-transitive, then G is of Lie type.
Two known flag-transitive GQ’s that are point-primitive butline-imprimitive:
GQ(3,5),
GQ of order (15,17) arising from Lunelli-Sce hyperoval.
3G has a unique minimal normal subgroup N, and N is a nonabelian simplegroup: N E G 6 Aut(N)
John Bamberg Symmetric Finite Generalised Polygons
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For generalised quadrangles
B., Giudici, Morris, Royle, Spiga (2012)If G acts primitively on the points and lines of a thick GQ then:
G is almost simple3.
If G is also flag-transitive, then G is of Lie type.
Two known flag-transitive GQ’s that are point-primitive butline-imprimitive:
GQ(3,5),
GQ of order (15,17) arising from Lunelli-Sce hyperoval.
3G has a unique minimal normal subgroup N, and N is a nonabelian simplegroup: N E G 6 Aut(N)
John Bamberg Symmetric Finite Generalised Polygons
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O’Nan-Scott in a nutshell
Theorem (The ‘O’Nan-Scott’ Theorem)
Suppose a finite permutation group G acts primitively on a set Ω.Then one of the following occurs:
K Abelian
HA
K simple
CG (K) = 1
K regular
AS
HS
CG (K) = 1
Kα subdirect K
Kα simple
PA
TW
HC SD
CD
John Bamberg Symmetric Finite Generalised Polygons
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O’Nan-Scott in a nutshell
Theorem (The ‘O’Nan-Scott’ Theorem)
Suppose a finite permutation group G acts primitively on a set Ω.Then one of the following occurs:
K Abelian
HA
K simple
CG (K) = 1
K regular
AS
HS
CG (K) = 1
Kα subdirect K
Kα simple
PA
TW
HC SD
CD
(HC) K .Inn(K) E G 6 K .Aut(K)Ω = K , holomorph action
(SD) K = T k E G 6 T k .(Out(T ) × Sk ),
Ω = T k−1, diagonal action
John Bamberg Symmetric Finite Generalised Polygons
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K Abelian
HA
K simple
CG (K) = 1
K regular
AS
HS
CG (K) = 1
Kα subdirect K
Kα simple
PA
TW
HC SD
CD
John Bamberg Symmetric Finite Generalised Polygons
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K Abelian
HAHA
K simple
CG (K) = 1
K regular
AS
HSHS
CG (K) = 1
Kα subdirect K
Kα simple
PA
TW
HCHC SD
CD
Point-primitive & line-transitive4
4B., Glasby, Popiel, Praeger 2017John Bamberg Symmetric Finite Generalised Polygons
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K Abelian
HAHAHA
K simple
CG (K) = 1
K regular
AS
HS
CG (K) = 1
Kα subdirect K
Kα simple
PA
TW
HCHC SD
CD
Point-primitive4
4B., Popiel, Praeger 2019John Bamberg Symmetric Finite Generalised Polygons
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K Abelian
HAHA
K simple
CG (K) = 1
K regular
ASAS
HSHS
CG (K) = 1
Kα subdirect K
Kα simple
PAPA
TWTW
HCHC SDSD
CDCD
Point-primitive (aim)
John Bamberg Symmetric Finite Generalised Polygons
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B., Popiel, Praeger (2019)
If G acts primitively on the points of a thick GQ, not affine, thenone of the following occurs:
type soc(G) necessary conditions
HS T × T T has Lie type with Lie rank 6 7SD T k T has Lie type with Lie rank 6 8,
or T = Altm with m 6 18, or T sporadicCD (T k)r r 6 3; T has Lie type with Lie rank 6 3,
or T = Altm with m 6 9, or T sporadicPA T r r 6 4;AS, TW – some information on fixities
Remark
With some extra work, we think HS can be removed completely.
John Bamberg Symmetric Finite Generalised Polygons
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Generalised hexagons and octagons
Schneider & Van Maldeghem (2008)A group acting flag-transitively, point-primitively andline-primitively on a generalised hexagon or octagon is almostsimple of Lie type.
B., Glasby, Popiel, Praeger, Schneider (2017)A group acting point-primitively on a generalised hexagon oroctagon is almost simple of Lie type.
Morgan & Popiel (2016)Moreover, if T 6 G 6 Aut(T ) with T simple, then
(i) T 6= 2B2(q) or 2G2(q);
(ii) if T = 2F4(q), then Γ is the classical generalised octagon orits dual.
John Bamberg Symmetric Finite Generalised Polygons
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Generalised hexagons and octagons
Schneider & Van Maldeghem (2008)A group acting flag-transitively, point-primitively andline-primitively on a generalised hexagon or octagon is almostsimple of Lie type.
B., Glasby, Popiel, Praeger, Schneider (2017)A group acting point-primitively on a generalised hexagon oroctagon is almost simple of Lie type.
Morgan & Popiel (2016)Moreover, if T 6 G 6 Aut(T ) with T simple, then
(i) T 6= 2B2(q) or 2G2(q);
(ii) if T = 2F4(q), then Γ is the classical generalised octagon orits dual.
John Bamberg Symmetric Finite Generalised Polygons
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Generalised hexagons and octagons
Schneider & Van Maldeghem (2008)A group acting flag-transitively, point-primitively andline-primitively on a generalised hexagon or octagon is almostsimple of Lie type.
B., Glasby, Popiel, Praeger, Schneider (2017)A group acting point-primitively on a generalised hexagon oroctagon is almost simple of Lie type.
Morgan & Popiel (2016)Moreover, if T 6 G 6 Aut(T ) with T simple, then
(i) T 6= 2B2(q) or 2G2(q);
(ii) if T = 2F4(q), then Γ is the classical generalised octagon orits dual.
John Bamberg Symmetric Finite Generalised Polygons
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Antiflag
Generalised quadrangle
Given an antiflag (P, `), there is a unique line m on P concurrentwith `.
John Bamberg Symmetric Finite Generalised Polygons
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Antiflag
Generalised quadrangle
Given an antiflag (P, `), there is a unique line m on P concurrentwith `.
John Bamberg Symmetric Finite Generalised Polygons
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Antiflag
Generalised quadrangle
Given an antiflag (P, `), there is a unique line m on P concurrentwith `.
John Bamberg Symmetric Finite Generalised Polygons
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Theorem (B., Li, Swartz 2018)
Let Q be a finite thick generalised quadrangle and supposeG 6 Aut(Q) acting transitively on the antiflags. Then Q isclassical or GQ(3, 5), GQ(5, 3).
strategy
G acts quasiprimitively on points OR lines.
G point-primitive & line-imprimitive =⇒ Q ∼= GQ(3, 5).
Reduce to G acting primitively on both points and lines of almost simple type.
|TP |3 > |T | where soc(G) = T ; use the characterisation result by Alavi andBurness to determine possibilities for G and GP .
John Bamberg Symmetric Finite Generalised Polygons
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Groups-on-graphs Geometry
Locally 3-arc transitive Antiflag transitiveLocally 2-arc transitive Transitive on collinear point-pairs and
concurrent line-pairsEdge transitive Flag transitive
John Bamberg Symmetric Finite Generalised Polygons
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Theorem (B., Li, Swartz (submitted))
If Q is a thick locally (G , 2)-transitive generalised quadrangle, thenone of the following holds:
Q ∼= GQ(3, 5),GQ(5, 3), or
Q is classical.
strategy
G acts quasiprimitively on points OR lines.
G point-quasiprimitive & line-nonquasiprimitive =⇒ Q ∼= GQ(3, 5).
Reduce to G acting primitively on points, almost simple type, socle of Lie type.
|TP |3 > |T | where soc(G) = T ; use the characterisation result by Alavi andBurness to determine possibilities for G and GP .
John Bamberg Symmetric Finite Generalised Polygons
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Open problems
1 Show that if G acts flag-transitive on a finite GQ, then G actsprimitively on points OR lines.
2 Are all point-primitive GQ’s point-distance-transitive?
3 Find new generalised hexagons and octagons.
4 Show that if G acts primitively on the points of a finite GQ,and intransitively on the lines, then the G -orbits on linesdivide them in half. (James Evans)
John Bamberg Symmetric Finite Generalised Polygons