Also available at http://amc.imfm.si
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ARS MATHEMATICA CONTEMPORANEA 3 (2010) 177–192
On the rank two geometries of the groups
PSL(2, q): part I
Julie De Saedeleer , Dimitri Leemans ∗
Université Libre de Bruxelles, Département de Mathématiques - C.P.216,
Boulevard du Triomphe, B-1050 Bruxelles, Belgium
Received 25 June 2009, accepted 30 June 2010, published online 10 November 2010
Abstract
We determine all firm and residually connected rank 2 geometries on which PSL(2, q)
acts flag-transitively, residually weakly primitively and locally two-transitively, where one
of the maximal parabolic subgroups is isomorphic to Eq :
(q−1)
(2,q−1) , where Eq denotes an
elementary abelian group of order q, or D2n(q), the dihedral group of order 2n(q) where
n(q) := (q±1)gcd(2,q−1) for some prime-power q.
Keywords: Projective special linear groups, coset geometries, locally s-arc-transitive graphs.
Math. Subj. Class.: 51E24, 05C25
1 Introduction
Following Tits’ geometric interpretation of the exceptional complex Lie groups [23, 25],
Francis Buekenhout generalized in [2] and [3] certain aspects of this theory in order to
achieve a combinatorial understanding of all finite simple groups. Since then, two main
approaches have been developed in diagram geometry. One is to classify geometries over
a given diagram, mainly geometries extending buildings (see for example [7], chapter 22
for a survey and [24] for the theory of buildings). Another approach is to classify coset
geometries for a given group under certain conditions. Rules for such classifications have
been stated by Buekenhout in [5] and [6].
Since 1993, several people, including Olivier Bauduin, Francis Buekenhout, Philippe
Cara, Michel Dehon, Maya Gailly, Harald Gottschalk, Xavier Miller, Koen Vanmeerbeek
and the authors, have classified geometries under the following assumptions. The geome-
tries obtained must be firm, residually connected, flag-transitive and residually weakly
primitive, and they must satisfy the intersection property of rank two. Moreover, since
∗Supported by the“Communauté Française de Belgique - Actions de Recherche Concertées
E-mail addresses: judesaed@ulb.ac.be (Julie De Saedeleer), dleemans@ulb.ac.be (Dimitri Leemans)
Copyright c© 2010 DMFA Slovenije
178 Ars Math. Contemp. 3 (2010) 177–192
2000, Buekenhout, Cara, Dehon and Leemans decided to add one further axiom: the ge-
ometries must also be locally two-transitive. For a detailed survey of this work, we refer
to [9]. Most of these classifications have been obtained using a series of CAYLEY or more
recently MAGMA programs (see [12] for the programs, and [1] for MAGMA). We refer
to [18] and its references for the most recent classifications related to the nine smallest
sporadic simple groups. For his doctorate thesis, Leemans classified in a series of papers
the residually weakly primitive geometries of the Suzuki simple groups (see [17] and its
references). That classification does not rely on computer algebra. The aim of this paper
is to start a similar classification for the groups PSL(2, q). We present the classification of
rank two coset geometries for the groups PSL(2, q), with q a power of a prime, satisfying
all of the above conditions but the intersection property. We choose to study these simple
groups because, after the Suzuki groups, they have the simplest subgroup structure and are
therefore the groups for which a theoretical classification of their geometries is more likely
to succeed. Moreover, these groups are embedded in lots of other simple groups, especially
in sporadic groups. We focus on the cases where one of the maximal parabolic subgroups
is isomorphic to Eq :
(q−1)
(2,q−1) (where (2, q − 1) denotes the greatest common divisor of 2
and q − 1) or D2n(q). The other cases will be analysed in a subsequent paper.
The paper is organised as follows. In Section 2, we recall the basic definitions and
notations needed in this paper. In Section 3, we give a sketch proof of our main result:
Theorem 1.1. LetG ∼= PSL(2, q) and Γ(G; {G0, G1, G0∩G1}) be a locally two-transitive
RWPRI coset geometry of rank two. If G0 is isomorphic to one of Eq :
(q−1)
(2,q−1) or D2n(q),
then Γ is isomorphic to one of the geometries appearing in Table 1.
The Buekenhout diagrams of the geometries are given in the statements of Propositions 5.3,
5.8 and 5.12. In Section 4, we recall the subgroup lattice of PSL(2, q), and we give the two-
transitive representations of the maximal subgroups. In Section 5, we prove Theorem 1.1,
which is based on the proof of Propositions 5.3, 5.8 and 5.12. For that, we determine the
rank 2 RWPRI and (2T )1 geometries of PSL(2, q) and their number, up to isomorphism
and up to conjugacy. We also determine their Buekenhout diagrams.
The geometries obtained in Theorem 1.1 are also locally 2-arc-transitive graphs in the
sense of [15]. These graphs are interesting in their own right because of the numerous
connections they have with other fields of mathematics (see [15] for more details). We
also refer to the classification of these graphs for almost simple groups with socle a Ree
simple group Ree(q) (see [14]). In terms of locally 2-arc-transitive graphs, we obtain
here the classification of these graphs with one vertex-stabilizer maximal in PSL(2, q) and
isomorphic to Eq :
(q−1)
(2,q−1) or D2n(q). The last column of Table 1 gives, for each geometry
Γ, the value of s such that Γ is a locally s-arc-transitive graph but not a locally (s+ 1)-arc-
transitive.
Acknowledgements : The authors would like to thank Francis Buekenhout, Cheryl E.
Praeger and the referee for fruitful comments on a preliminary version of this paper.
2 Definitions and notations
In this section, we recall the basic notions on coset geometries and give the definitions
needed to understand this paper.
J. De Saedeleer and D. Leemans: On the rank two geometries of the groups. . . 179
G0 ∼= Eq : (q−1)(2,q−1) q = p
n,
with p prime
G0 ∩G1 G1 ] Geom. ] Geom. Extra condi- locally(G, s)-arc-
up to conj. up to isom. tions on q transitive graphs
(q − 1) Eq : (q − 1) 1 1 q = 2n s = 2
(q − 1) D2(q−1) 1 1 q = 2n s = 3
G0 ∼= D2 q−1
(2,q−1)
q = pn,
with p prime
G0 ∩G1 G1 ] Geom. ] Geom. Extra condi- locally(G, s)-arc-
up to conj. up to isom. tions on q transitive graphs
(q − 1) Eq : (q − 1) 1 1 q = 2n s = 3
2 S3 1 1 q = 4 s = 2
2 22 1 1 q = 4 s = 3
22 D12 2 1 q = 13 s = 3
D8 S4 2 1 q = 17 s = 7
D8 S4 2 1 q = 25 s = 3
D10 A5 1 1 q = 16 s = 3
D10 A5 2 1 q = 31 s = 3
G0 ∼= D2 q+1
(2,q+1)
q = pn,
with p prime
G0 ∩G1 G1 ] Geom. ] Geom. Extra condi- locally(G, s)-arc-
up to conj. up to isom. tions on q transitive graphs
2 S3 1 1 q = 5 s = 2
2 22 1 1 q = 5 s = 3
3 A4 1 1 q = 5 s = 3
22 D12 2 1 q = 11 s = 3
D6 S4 2 1 q = 17 s = 3
D8 S4 2 1 q = 23 s = 3
D10 A5 2 1 q = 19, 29 s = 3
Table 1: The RWPRI and (2T )1 geometries with G0 ∼= Eq : (q−1)(2,q−1) or D2n(q).
180 Ars Math. Contemp. 3 (2010) 177–192
2.1 Coset geometries, their properties and Buekenhout diagram
A general reference for diagram geometries and their properties is [7]. Let I = {1, . . . , n}
be a finite set, called the type set. Its elements are called types. Let G be a group and
(Gi)i∈I be a collection of distinct subgroups ofG. LetX := {Gig : g ∈ G,Gi ∈ (Gi)i∈I}
be the set of their cosets. We define a pregeometry Γ = Γ(G; (Gi)i∈I) = (X, ∗, t) provided
with a type function t : Gig 7→ i and an incidence relation ∗ ⊂ X ×X , such that
Gig ∗Gjh if and only if Gig ∩Gjh 6= ∅.
The number n = |I| is called the rank of Γ. A flag F of Γ is a set of pairwise incident
elements, and t(F) := {t(x) : x ∈ F} is called its type. A flag C with t(C) = I is called
a chamber. If every flag of Γ is contained in a chamber, Γ is called a (coset) geometry. A
geometry is firm (resp. thin, thick) if any flag which is not a chamber is contained in at least
two (resp. exactly two, at least three) chambers.
The residue of a flag F of Γ is the geometry ΓF consisting of the elements of Γ\F
incident with all elements of F , together with the restricted type-function and induced
incidence relation. Let F be a flag of type J ⊂ I . Then ΓF is a geometry over the typeset
I − J . A coset geometry Γ is called residually connected if the incidence graph of every
residue of rank at least two is connected. For any J ⊂ I , we setGJ :=
⋂
j∈J Gj ,B := GI .
We call L(Γ) := {GJ : J ⊂ I} the sublattice (of the subgroup lattice of G) spanned by
the collection (Gi)i∈I . The group B is said to be the Borel subgroup of L(Γ). We say
that L(Γ) is strongly boolean if, for any two elements of L(Γ), their lowest upper bound
in L(Γ) is the subgroup that they generate in G. The following results is used to check
whether a pregeometry Γ is a residually connected geometry.
Lemma 2.1. [24] Let Γ = Γ(G; (Gi)i∈I) be a pregeometry. Then Γ is a residually con-
nected pregeometry if and only if L(Γ) is strongly boolean.
We callG flag-transitive ifG acts transitively on the set of flags of type J for all subsets
J of I .
In this paper, we are interested only in rank two geometries. The following lemma is
obvious.
Lemma 2.2. Let Γ = Γ(G; {G0, G1}) be a pregeometry of rank two. Then Γ is a geometry
and G acts flag-transitively on it.
We call Γ primitive provided that G acts primitively on the set of i-elements of Γ, for
each i ∈ I . Moreover, we call Γ residually primitive (RPRI) if each residue ΓF of a flag
F is primitive for the group induced on ΓF by the stabilizer GF of F . We say that Γ is
weakly primitive (WPRI) if Gi is maximal in G for at least one i ∈ I . Moreover, Γ is said
to be residually weakly primitive (RWPRI) provided that ΓF is WPRI for every flag F .
We say that Γ satisfies the intersection property of rank two (IP )2 if every residue of
rank two is either a partial linear space or a generalized digon. Note that this condition
excludes all 2− (v, k, λ) designs, λ ≥ 2, except the generalized digons.
We call Γ locally 2-transitive and we write (2T )1 for this, provided that the stabilizer
GF of any flag F of rank n− 1 acts 2-transitively on the residue ΓF .
Following [2], the Buekenhout diagram of a firm, residually connected, flag-transitive
geometry Γ is a graph together with additional structure, whose vertices are the elements
of I , which is further described as follows. To each vertex i ∈ I , we attach the order si that
J. De Saedeleer and D. Leemans: On the rank two geometries of the groups. . . 181
is | ΓF | −1, where F is any flag of type I\{i}, the number ni of varieties of type i, that
is the index of Gi in G, and the subgroup Gi. Elements i, j of I are not joined by an edge
of the diagram provided a residue ΓF of type {i, j} is a generalized digon. Otherwise, i
and j are joined by an edge endowed with three positive integers dij , gij , dji, where gij(the
gonality) is equal to half the girth of the incidence graph of a residue ΓF of type {i, j} and
dij (resp. dji), the i-diameter (resp. j-diameter) is the greatest distance from some fixed
i-element (resp. j-element) to any other element in the incidence graph of ΓF . On a picture
of the diagam, this structure will often be depicted as follows.
i idij gij dji
si sj
Ni Nj
Gi Gj
Borel = Gi ∩Gj
If gij = dij = dji = n, then ΓF is called a generalized n-gon and we do not write dij and
dji. If dij = gij = 3, dji = 4 and si = 1, we write c instead of gij dij dji.
We classify the rank two geometries Γ = Γ(G; {G0, G1}) for G = PSL(2, q), with q a
power of a prime, under the following conditions. The geometry Γ must be firm, residually
connected, and the group G must act flag-transitively and residually weakly primitively on
Γ. Moreover, Γ must be locally two-transitive. Our search for such geometries is based on
the following obvious lemma.
Lemma 2.3. If Γ = Γ(G; {G0, G1}) is residually weakly primitive, then Γ is firm and
residually connected.
Hence Lemmas 2.2 and 2.3 imply that we only have to test the (RWPRI) and (2T )1
conditions.
Let G be a group and Aut(G) be its automorphism group. The coset geometries
Γ(G; {G0, G1}) and Γ(G; {G′0, G′1}) are conjugate (resp. isomorphic) provided there
exists an element g ∈ G (resp. g ∈ Aut(G)) such that {Gg0, G
g
1} = {G′0, G′1} (resp.
{g(G0), g(G1)} = {G′0, G′1}). We classify geometries up to conjugacy and up to iso-
morphism. That is, for each triple {G0, G1, G0 ∩ G1}, we give the number of classes of
geometries with respect to conjugacy and isomorphism.
From now on, we denote Γ = Γ(G; {G0, G1}) by Γ(G;G0, G1, G0 ∩G1) and we say
this a coset geometry of rank two.
Throughout this paper, we use the notation of the ATLAS [11] for groups.
2.2 Locally s-arc-transitive graphs
The following definitions are taken from [15]. Let G be a finite simple undirected con-
nected graph. Denote by V (resp. E) its vertex-set (resp. edge-set). The edge-set may be
identified with a subset of unordered pairs of elements of V . An s-arc is an ordered (n+1)-
tuple (α0, ..., αn) of vertices such that {αi−1, αi} is an edge of G for all i = 1, ..., n and
αj−1 6= αj+1 for all j = 1, ..., n − 1. Let G be a subgroup of the automorphism group
Aut(G) of G. The graph G is said to be (G, s)-arc-transitive if G is transitive on the set
of s-arcs of G; also, G is said to be s-arc-transitive if it is (Aut(G), s)-arc-transitive. Sim-
ilarly G is said to be (G, 1)-arc-transitive if G is transitive on the 1-arcs of G, that is on
182 Ars Math. Contemp. 3 (2010) 177–192
the ordered pairs (α0, α1) where {α0, α1} is an edge of G. Given G ≤ Aut(G), we call G
locally (G, s)-arc-transitive if G contains an s-arc and given any two s-arc α and β starting
at the same vertex v, there exists an element g ∈ Gv mapping α to β. We say G is locally
s-arc-transitive if it is locally (G, s)-arc-transitive for some G ≤ Aut(G).
Obviously, 1-arc transitivity is equivalent to flag-transitivity if the graph G is seen as a
rank two geometry whose elements of type 0 (resp. 1) are the vertices (resp. edges) of G.
Moreover, local 2-arc transitivity is equivalent to property (2T )1 defined in the previous
section.
3 Sketch of the proof of Theorem 1.1
Let G be a group. Let G0 and G1 be subgroups of G and let G01 := G0 ∩G1. The RWPRI
condition in rank two requires that either G0 or G1 is a maximal subgroup of G and that
G01 is a maximal subgroup of G0 and G1. The (2T )1 condition requires that both G0
and G1 act two-transitively on the respective cosets of G01. We use the following method
to determine all the RWPRI and (2T )1 geometries of the projective special linear group
PSL(2, q). We choose a subgroup G0 and we insist that it must be a maximal subgroup
of PSL(2, q). Then we determine all the possibilities for G01. They are the maximal
subgroups of G0 such that the action of G0 on the cosets of G01 in G0 is two-transitive.
Finally, we look at all the subgroups G1 which contain G01 as a maximal subgroup and act
two-transitively on the cosets of G01 in G1. Therefore, to achieve our goal, we first must
determine what are the possibilities for G0. These are the maximal subgroups of G. They
are given in the next section, as well as their maximal subgroups, the possibilities for G01.
In Section 5, we then determine what are the possibilities for G1.
4 Structure of subgroups of PSL(2, q)
To follow the approach described above, we first recall the list of subgroups of the pro-
jective special linear groups PSL(2, q). We then give the list of maximal subgroups of
PSL(2, q) and the list of the maximal subgroups of each maximal subgroup of PSL(2, q).
Finally, in order to be able to check the (2T )1 property, we determine the two-transitive
representations of the maximal subgroups of PSL(2, q).
4.1 The subgroups of PSL(2, q)
The subgroup structure of PSL(2, q) may be found in Dickson [13] or Huppert [16]. It was
first obtained in papers by Moore [20] and Wiman [27].
Lemma 4.1. [Dickson] The group PSL(2, q) of order q(q
2−1)
(2,q−1) , where q = p
n (p prime),
contains only the following subgroups:
1. q + 1 elementary conjugate abelian subgroups of order q, denoted by Eq .
2. q(q±1)2 cyclic conjugate subgroups of order
(q±1)
(2,q−1) , denoted by
(q±1)
(2,q−1) .
3. q(q±1)2 cyclic conjugate subgroups of order d for every divisor d of
(q±1)
(2,q−1) , denoted
by d.
4. q(q
2−1)
2d(2,q−1) dihedral groups of order 2d, denoted by D2d for every d > 2 dividing
n(q) := (q±1)(2,q−1) . The number of conjugacy classes of these subgroups is one or two,
J. De Saedeleer and D. Leemans: On the rank two geometries of the groups. . . 183
depending on whether (q±1)d(2,q−1) is odd or even.
5. For q odd, q(q
2−1)
12(2,q−1) dihedral groups of order 4 denoted by 2
2. The number of conju-
gacy classes of these groups is one if q ≡ ±3(8) and two if q ≡ ±1(8). If q is even,
the groups 22 are in the family 6.
6. There are sets of q
2−1
(2,1,1)(pk−1) conjugate elementary abelian subgroups of order p
m,
denoted by Epm for all natural number m, such that 1 ≤ m ≤ n − 1, where k is a
common divisor of n and m and (2, 1, 1) is equal to 2 (resp. 1, 1) if p > 2 and nk is
even (resp. p > 2 and nk is odd, p = 2).
7. There are sets of (q
2−1)pn−m
(2,1,1)(pk−1) conjugate subgroups Epm : d which are semidirect
products of an elementary abelian group Epm and a cyclic group of order d, for
every natural number m such that 1 ≤ m ≤ n and every natural number d dividing
pk−1
(1,2,1) , where k is a common divisor of n and m and (1, 2, 1) is equal to 1 (resp. 2,
1) if p > 2 and nk is even (resp. p > 2 and
n
k is odd, p = 2).
8. For q odd or q = 4m, q(q
2−1)
12(2,q−1) subgroups isomorphic to A4. The number of conju-
gacy classes of these groups is one if q ≡ ±3(8) or q even and two if q ≡ ±1(8).
9. For q ≡ ±1(8), two classes of q(q
2−1)
24(2,q−1) symmetric groups S4, of order 24.
10. For q ≡ ±1(5), two classes of q(q
2−1)
60(2,q−1) alternating groups A5 (of order 60) and for
q = 4m one class of q(q
2−1)
60(2,q−1) alternating groups A5.
11. q(q
2−1)
pw(p2w−1) groups PSL(2, p
w) for every divisor w of r. The number of conjugacy
classes of these groups is two (resp. one) if p > 2 and rw is even (resp. p > 2 and
r
w
is odd, p = 2).
12. For q odd, two classes of q(q
2−1)
2pw(p2w−1) groups PGL(2, p
w) for every even divisor 2w
of r.
Remark 4.2. A5 subgroups are given either by case 10 (when q ≡ ±1(5) and q = 4m) or
by case 11 (when q ≡ 0(5)) of Lemma 4.1. Also, case 12 is a subcase of case 11 provided
q is even.
Remark 4.3. In the cases 6 and 7 of Lemma 4.1, the numbers of conjugacy classes are not
given. The number of conjugacy classes of the elementary abelian subgroups Epm given
by Dickson (see [13], §260) is wrong. Take for instance PSL(2, 64) and the subgroups iso-
morphic to E16. Dickson’s result gives 651 classes of such subgroups. Using MAGMA [1]
we find there are 11 conjugacy classes of subgroups E16. Ten classes are of length 4095
and one is of length 1365, the lengths given by Dickson.
Notice that Dickson does not give the number of conjugacy classes of the subgroups
Epm : d, except in the particular case where m = n and d = p
n−1
(2,q−1) . There are q + 1
subgroups Eq : q−1(2,q−1) , all conjugate.
4.2 Maximal subgroups of PSL(2, q)
Table 2 and 3 give lists of the maximal subgroups of PSL(2, q) in the case where q is even
or odd respectively. This result is known; see for example [22].
184 Ars Math. Contemp. 3 (2010) 177–192
Structure Order Index
Eq : (q − 1) q(q − 1) q + 1
D2(q+1) 2(q + 1)
q(q−1)
2
q 6= 2
D2(q−1) 2(q − 1) q(q+1)2
A5 60
q(q2−1)
60
q = 4r and r is a prime
PSL(2, q′) ∼= PGL(2, q′) q′(q′2 − 1) q(q
2−1)
q′(q′2−1)
q′ > 2, q = q′m, m is prime
or q′ = 2, q = q′2
Table 2: The maximal subgroups of PSL(2, q), for q even
Structure Order Index
Eq :
q−1
2
q(q−1)
2 q + 1
D(q+1) q + 1 q(q − 1)
q 6= 7, 9
D(q−1) q − 1 q(q+1)2
q 6= 3, 5, 7, 9, 11
A4 12
q(q2−1)
12×2
if q = p > 3 and q ≡ 3, 13, 27, 37(40) or q = 5
S4 24
q(q2−1)
24×2
if q = p > 2 and q ≡ ±1(8)
A5 60
q(q2−1)
60×2
if
 q = 5
r r odd prime or
p = q ≡ ±1(5) p prime or
q = p2 ≡ −1(5) p prime
PSL(2, q′) q
′(q′2−1)
2
q(q2−1)
q′(q′2−1)
q′ > 2, q = q′m, m odd prime
PGL(2, q′) q′(q′2 − 1) q(q
2−1)
q′(q′2−1)
q = q′2
Table 3: The maximal subgroups of PSL(2, q), for q odd
J. De Saedeleer and D. Leemans: On the rank two geometries of the groups. . . 185
G0 G01
Eq :
q−1
(2,q−1) Eq :
q−1
4 for q ≡ 1(4)
cyclic group of order (q − 1) for q even
D2n D2m if n = 2m or n = 3m; if n is prime D2 = C2
cyclic group of order n
A4 C3
S4 D6
D8
A4
A5 D10
A4
PSL(2, q) Eq :
q−1
(2,q−1)
C3 if q = 2
S4 if q = 7
A5 if q = 9, 11
PGL(2, q) Eq : (q − 1)
PSL(2, q) for q odd
Table 4: The two-transitive actions of the maximal subgroups of PSL(2, q)
4.3 Two-transitive representations of the maximal subgroups of PSL(2, q)
The first lemma is obvious but will be used often in the next section as a necessary condition
to have a two-transitive action.
Lemma 4.4. Let G be a group and let H be a subgroup of G. If G acts 2-transitively on
the cosets of H in G, then |G| must be divisible by [G :H]([G :H]− 1).
A group G is said to act regularly on a set Ω if G is transitive on Ω and the stabilizer in
G of a point p ∈ Ω is the identity.
Lemma 4.5. [26] Let (G,Ω) be a permutation group which is transitive over Ω and let G
be abelian. Then G is regular. Moreover, if G is 2-transitive then |Ω| = 2.
We now provide the classification (existence and uniqueness) all two-transitive repre-
sentations of every maximal subgroup of PSL(2, q), result borrowed from [10].
Lemma 4.6. Let G0 be a maximal subgroup of PSL(2, q). Then G0 acts two-transitively
on the cosets of the subgroups G01 given in Table 4.
5 Proof of Theorem 1.1
In this section, we determine the rank 2 RWPRI and (2T )1 geometries of PSL(2, q). We
break down the task by classifying those geometries with a fixed subgroup G0. Since
we may assume without loss of generality that G0 is maximal in G, we follow Tables 2
and 3 that give all maximal subgroups of PSL(2, q). The number of RWPRI and (2T )1
geometries of rank 2 depends on the value of q = pn. More precisely, it usually depends
186 Ars Math. Contemp. 3 (2010) 177–192
G0 ∼= Eq : (q−1)(2,q−1) q = p
n,
with p prime
G01 G1 ] Geom. ] Geom. Extra conditions
up to conj. up to isom. on q
Γ1 (q − 1) Eq : (q − 1) 1 1 q = 2n
Γ2 (q − 1) D2(q−1) 1 1 q = 2n
Table 5: The RWPRI and (2T )1 geometries with G0 ∼= Eq : (q−1)(2,q−1) .
on whether q is even or odd. Knowing that q = pn with p prime, the two cases are q = 2n
or q odd.
5.1 The case where G0 ∼= Eq : (q−1)(2,q−1)
By Lemma 4.6 the possibilities for G01 are Eq : q−14 provided q ≡ 1(4) and the cyclic
group of order (q − 1) provided q is even.
Lemma 5.1. Let G ∼= PSL(2, q). If q = 1(4) and H is a subgroup of G such that H
contains a subgroup isomorphic to Eq : q−14 maximally, and acts two-transitively on its
cosets. Then H ∼= Eq : (q−1)2 .
Proof. Left to the reader.
Lemma 5.2. Let G ∼= PSL(2, q). If q is even and H is a subgroup of G such that H
contains a subgroup isomorphic to a cyclic group of order (q − 1) maximally, and acts
two-transitively on its cosets. Then H ∼= Eq : (q − 1) or H ∼= D2(q−1).
Proof. Left to the reader.
Proposition 5.3. Let G ∼= PSL(2, q). Every RWPRI and (2T )1 geometry of rank two
Γ(G;G0, G1, G0 ∩G1) in which G0 ∼= Eq : (q−1)(2,q−1) is isomorphic to one of the geometries
appearing in Table 5. Their Buekenhout diagrams are as follows:
Γ1
g g3 2 3
q − 1 q − 1
q + 1 q + 1
Eq : (q − 1) Eq : (q − 1)
B = (q − 1)
RPRI
Γ2
g gc
1 q − 1
q + 1 q(q + 1)/2
Eq : (q − 1) D2(q−1)
B = (q − 1)
RPRI
Moreover, Γ1 is a Doubling of Γ2 and it does not satisfy the (IP )2 condition.
Proof. Let G0 ∼= Eq : (q−1)(2,q−1) . Lemma 4.6 gives two possibilities for G01.
Subcase 1: G01 := G0 ∩G1 ∼= Eq : q−14 provided q ≡ 1(4).
By Lemma 5.1 the only possibility for G1 is Eq : q−12 . We must still check whether
this geometry exists, that is whether there are two subgroups isomorphic to Eq : q−12 in
J. De Saedeleer and D. Leemans: On the rank two geometries of the groups. . . 187
PSL(2, q) that have a subgroup Eq : q−14 in common. Since PSL(2, q) is simple and
Eq :
q−1
2 maximal, Eq :
q−1
2 is self-normalized. Moreover, the group Eq :
q−1
2 contains
exactly one maximal subgroup of order Eq : q−14 . The normalizer of Eq :
q−1
4 in PSL(2, q)
is Eq : q−12 . Therefore the number of subgroups Eq :
q−1
2 containing a given subgroup
Eq :
q−1
4 in PSL(2, q) is equal to
| PSL(2, q) |
| Eq : q−12 |
· 1 ·
| Eq : q−12 |
| PSL(2, q) |
= 1.
Thus the geometry Γ
(
PSL(2, q);Eq :
q−1
2 , Eq :
q−1
2 , Eq :
q−1
4
)
does not exist.
Subcase 2: G01 := G0 ∩G1 ∼= (q − 1) provided q is even.
By Lemma 5.2 the possibilities for G1 are Eq : (q − 1) and D2(q−1).
Consider first the case where G1 ∼= Eq : (q− 1). We must check whether this geometry
exists, that is, if there are, in PSL(2, q), two subgroups isomorphic to Eq : (q− 1) that have
a cyclic group of order (q − 1) in common. Since PSL(2, q) is simple and Eq : (q − 1)
maximal,Eq : (q−1) is self-normalized. There is only one conjugacy class ofEq : (q−1) in
PSL(2, q). Moreover, the group Eq : (q−1) contains q maximal subgroups of order (q−1)
which are all conjugate in Eq : (q − 1). Therefore the number of subgroups Eq : (q − 1)
containing a given cyclic subgroup of order (q − 1) in PSL(2, q) is equal to
| PSL(2, q) |
| Eq : (q − 1) |
· q ·
| D2(q−1) |
| PSL(2, q) |
= 2,
and thus the geometry exists. Up to conjugacy there is exactly one RWPRI and (2T )1
geometry Γ1 := Γ(PSL(2, q);Eq : (q − 1), Eq : (q − 1), q − 1), and thus also exactly one
up to isomorphism.
Next, let us consider the case where G1 ∼= D2(q−1). There is only one conjugacy class
ofEq : (q−1) and also one ofD2(q−1) in PSL(2, q). Since PSL(2, q) is simple andD2(q−1)
maximal, D2(q−1) is self-normalized. Moreover, the group D2(q−1) contains one maximal
subgroup of order (q−1). The normalizer of a cyclic group of order (q−1) in PSL(2, q) is
a dihedral group D2(q−1). Therefore the number of subgroups D2(q−1) containing a given
cyclic subgroup of order (q − 1) in PSL(2, q) is equal to
| PSL(2, q) |
| D2(q−1) |
· 1 ·
| D2(q−1) |
| PSL(2, q) |
= 1
and thus the geometry exists. Up to conjugacy there exists exactly one RWPRI and (2T )1
geometry Γ2 := Γ(PSL(2, q);Eq : (q − 1), D2(q−1), q − 1) and thus also exactly one up to
isomorphism.
Elements of type 0 in Γ2 are the points of the projective line PG(1, q). Elements of
type 1 are the pairs of points of PG(1, q). Incidence is symmetrized inclusion. Hence,
this geometry is clearly a complete graph and its Buekenhout diagram follows. Applying
Corollary 4.1 of [19] to Γ2, we get Γ1 and the corresponding Buekenhout diagram (see [19],
Table 1 or [21]). The (IP )2 condition is clearly satisfied in Γ2 and not in Γ1.
Observe that the geometries given in the previous theorem also satisfy the RPRI condi-
tion since D2(q−1) is a maximal subgroup of PSL(2, q).
188 Ars Math. Contemp. 3 (2010) 177–192
5.2 The case where G0 ∼= D2 q−1(2,q−1)
Recall that following Table 3 if q is odd, then q > 11. By Lemma 4.6, the possible
subgroups for G01 are D2d with q−1(2,q−1) = 2d or 3d and the cyclic group of order
q−1
(2,q−1) .
For each of these G01 we look for the various possible groups G1 in one of the following
four lemmas.
Lemma 5.4. Let G ∼= PSL(2, q). If q is odd, q > 11 and H is a subgroup of G such
that H contains a subgroup isomorphic to a cyclic group of order q−12 maximally, and acts
two-transitively on its cosets. Then H ∼= Dq−1.
Proof. Left to the reader.
When q is even, we distinguish the case q = 4 for clarity.
Lemma 5.5. LetG ∼= PSL(2, 4). IfH is a subgroup ofG such thatH contains a subgroup
isomorphic to a C2 maximally, and acts two-transitively on its cosets. Then H ∼= S3 or
H ∼= 22.
Proof. Straightforward.
Lemma 5.6. Let G ∼= PSL(2, q). If q = 2n > 2, q = 1(3) and H is a subgroup of G such
that H contains a subgroup isomorphic to D
2
(q−1)
3
maximally, and acts two-transitively on
its cosets. Then H ∼= D2(q−1) or H ∼= A5 provided q = 16.
Proof. Left to the reader.
Lemma 5.7. Assume q is odd and q > 11 and let G ∼= PSL(2, q).
1. If q = 1(4) and H is a subgroup of G such that H contains a subgroup isomorphic
to D
2
(q−1)
4
maximally, and acts two-transitively on its cosets. Then H ∼= Dq−1, or
H ∼= S4 provided q = 17.
2. If q = 1(6) and H is a subgroup of G such that H contains a subgroup isomorphic
to D
2
(q−1)
6
maximally, and acts two-transitively on its cosets. Then H ∼= Dq−1, or
H ∼= S4 provided q = 25 or H ∼= A5 provided q = 31.
Proof. Left to the reader.
The proof of the following Proposition is very similar to the one for Proposition 5.3.
Therefore we do not give the details.
Proposition 5.8. Let G ∼= PSL(2, q) with q 6= 3, 5, 7, 9 or 11. Every RWPRI and (2T )1
geometry of rank two Γ(G;G0, G1, G0 ∩ G1) in which G0 ∼= D2 (q−1)
(2,q−1)
is isomorphic to
one of the geometries appearing in Table 6. Their Buekenhout diagrams are as follows:
Γ1
g gc
1 q − 1
q + 1 q(q − 1)/2
Eq : (q − 1) D2(q−1)
B = (q − 1)
RPRI
Γ2
g g5 3 5
2 2
10 10
S3 S3
B = C2
RPRI
Desargues configuration
(doubling of the Petersen graph)
J. De Saedeleer and D. Leemans: On the rank two geometries of the groups. . . 189
G0 ∼= D2 q−1
(2,q−1)
q = pn,
with p prime
G01 G1 ] Geom. ] Geom. Extra conditions
up to conj. up to isom. on q
Γ1 (q − 1) Eq : (q − 1) 1 1 q = 2n
Γ2 2 S3 1 1 q = 4
Γ3 2 2
2 1 1 q = 4
Γ4 2
2 D12 2 1 q = 13
Γ5,Γ6 D8 S4 2 1 q = 17, 25
Γ7 D10 A5 1 1 q = 16
Γ8 D10 A5 2 1 q = 31
Table 6: The RWPRI and (2T )1 geometries with G0 ∼= D2 q−1
(2,q−1)
.
Γ3
g g5 5 6
1 2
10 15
S3 2
2
B = C2
Petersen graph
g gΓ4 9 6 9
2 2
91 91
D12 D12
B = 22
RPRI
Γ5
g g14 9 14
1 2
102 136
S4 D16
B = D8
RPRI
Γ6
g g13 5 13
2 2
325 325
D24 S4
B = D8
Γ7
g g7 4 7
2 5
68 136
A5 D30
B = D10
RPRI
Γ8
g g8 4 7
5 2
496 248
D30 A5
B = D10
RPRI
Observe that Γ1 is one of the two geometries found in section 5.2. Geometries Γ2, Γ4,
Γ5, Γ6 with q = 17 and Γ7 are in [8]. For Γ3, see [4]. To the best of our knowledge Γ6
with q = 25 and Γ8 are new.
5.3 The case where G0 ∼= D2 q+1(2,q+1)
Recall that following Table 2 and Table 3 the subgroup D2 q+1
(2,q−1)
is maximal in G if q 6=
2, 7, 9.
By Lemma 4.6, the possible subgroups for G01 are D2d with q+1(2,q+1) = 2d or 3d, and
the cyclic group of order q+1(2,q+1) . For each of these G01 we look for the various possible
groups G1 in one of the following three lemmas.
Lemma 5.9. Let G ∼= PSL(2, q) and assume q 6= 2, 7, 9 as required. If H is a subgroup
of G such that H contains a subgroup isomorphic to a cyclic group of order q+1(2,q+1) max-
190 Ars Math. Contemp. 3 (2010) 177–192
imally, and acts two-transitively on its cosets. Then H ∼= D2 q+1
(2,q−1)
or H ∼= A4 provided
q = 5.
Proof. Left to the reader.
Lemma 5.10. Let G ∼= PSL(2, q). If q 6= 2, q is even, q = −1(3) and H is a subgroup
of G such that H contains a subgroup isomorphic to D2( q+13 ) maximally, and acts two-
transitively on its cosets. Then H ∼= D2(q+1).
Proof. Left to the reader.
Lemma 5.11. Let G ∼= PSL(2, q) with q odd and q 6= 7, 9.
1. If q = −1(4) and H is a subgroup of G such that H contains a given subgroup
isomorphic to D
2
(q+1)
4
maximally and acts two-transitively on its cosets. Then H ∼=
Dq+1 or H ∼= A5 provided q = 19.
2. If q = −1(6) and H is a subgroup of G such that H contains a given subgroup
isomorphic to D
2
(q+1)
6
maximally, and acts two-transitively on its cosets. Then H ∼=
Dq+1, or H ∼= S4 provided q = 17, 23 or H ∼= A5 provided q = 29 or H ∼= 22
provided q = 5.
Proof. Left to the reader.
The proof of the following Proposition is very similar to the one of Proposition 5.3.
Therefore we do not give the details.
Proposition 5.12. Let G ∼= PSL(2, q) with q 6= 2, 7, 9. Every RWPRI and (2T )1 geometry
of rank two Γ(G;G0, G1, G0 ∩ G1) in which G0 ∼= D2 (q+1)
(2,q−1)
is isomorphic to one of the
geometries appearing in Table 7. Their Buekenhout diagrams are as follows:
Γ3
g gc
1 3
5 10
A4 S3
B = C3
RPRI
Γ4
g g7 5 7
2 2
55 55
D12 D12
B = 22
RPRI
Γ5
g g8 4 8
2 3
102 136
S4 D18
B = S3
RPRI
Γ6
g g11 6 11
2 2
253 253
D24 S4
B = D8
RPRI
Γ7
g g8 5 7
5 1
171 57
D20 A5
B = D10
RPRI
Γ8
g g8 4 7
5 2
406 203
D30 A5
B = D10
RPRI
J. De Saedeleer and D. Leemans: On the rank two geometries of the groups. . . 191
G0 ∼= D2 q+1
(2,q+1)
q = pn,
with p prime
G01 G1 ] Geom. ] Geom. Extra conditions
up to conj. up to isom. on q
Γ1 C2 S3 1 1 q = 5
Γ2 C2 2
2 1 1 q = 5
Γ3 C3 A4 1 1 q = 5
Γ4 2
2 D12 2 1 q = 11
Γ5 D6 S4 2 1 q = 17
Γ6 D8 S4 2 1 q = 23
Γ7, Γ8 D10 A5 2 1 q = 19, 29
Table 7: The RWPRI and (2T )1 geometries with G0 ∼= D2 q+1
(2,q+1)
.
Observe that Γ3 is a special case of the second geometry of Proposition 5.3 provided
q = 4. Moreover, Γ1 and Γ2 are also the same as geometries obtained in section 5.2 since
PSL(2, 4) ∼= PSL(2, 5). For Γ1, Γ3, Γ4, Γ5 and Γ8 with q = 19, see [8], and for Γ2 and
Γ7 see [4]. Observe that Γ7 is a truncation of Coxeter’s 57-cells. It is its vertex-edge graph
also called the Perkel graph. To the best of our knowledge Γ6 and Γ8 provided q = 29 are
new.
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