Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 39 (1998), No. 1, 97-120.

The Rank 2 Geometries of the Simple Suzuki Groups Sz(q)

Dimitri Leemans

Université Libre de Bruxelles, Département de Mathématiques - C.P.216 Boulevard du Triomphe, B-1050 Bruxelles e-mail: dleemans@cso.ulb.ac.be

Abstract. We determine all firm and residually connected rank 2 geometries on which a Suzuki simple group Sz(q) acts flag-transitively and residually weakly primitively.

1. Introduction

In 1986, Francis Buekenhout advocated in [1] a systematic search of the geometries to be obtained from a group and some of its subgroups.

In 1994, Michel Dehon presented in [10] a set of CAYLEY programs in order to classify all firm, residually connected and flag-transitive geometries for a given group G with the additional restriction that each stabilizer of some element is a maximal subgroup of G. Several groups were investigated as for example, U_{4}(2) [10], M_{11} [6], etc. An atlas was also made, containing a collection of projective groups P SL(2, q) [4].

This experimental work led to new rules for such classifications. In 1993, Francis Buekenhout and Michel Dehon changed the restriction on the subgroups forming the geometries, taking a residually weakly primitive condition. Again, experimental work was accomplished in that way. An atlas of residually weakly primitive geometries for small groups was achieved [5], and Harald Gottschalk determined all the geometries of the group P SL(3, 4) satisfying these conditions in his Diplomarbeit [13]. More recently, Dehon and Miller determined in [11] all geometries of M_{11} satisfying these new conditions and Gottschalk and Leemans classified the geometries of J_{1} [12].

From now on, it really seems that the residually weakly primitive restriction on the subgroups is a “good one”.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1998 Heldermann Verlag

98 D. Leemans: The rank 2 geometries of the simple Suzuki groups Sz(q)

The experience that we acquired leads us to be more ambitious. We now look at an infinite class of groups, and want to classify all geometries satisfying some conditions for all the groups of this class. We choose to study the family of the simple Suzuki groups. This choice is motivated by the well known fact that the structure of the Suzuki groups is particularly easy, compared to the other simple groups. Using the same conditions as in [5], except for the intersection property, we classify in this article all the firm, residually connected, flag-transitive and residually weakly primitive geometries of rank 2 that can be obtained from a Suzuki group Sz(q) where q = 2^{2e+1} and e is at least 1.

For a history of the Suzuki groups, we refer to [3].

The paper is organized as follows.

In Section 2, we recall definitions and notation for incidence geometry and the Suzuki groups, and we explain some notation we use throughout this paper. In Section 3, we give some properties on certain numbers that play an important role in our classification. In Section 4, we recall the structure of maximal subgroups of a Suzuki group. In Section 5, we give the structures of the maximal subgroups of each maximal subgroup of a Suzuki group. In Section 6, we determine all rank 2 geometries that are obtained. The results of the quite long discussion and case analysis of section 6 leads to a classification theorem that we state in Section 7. We give in Section 8 the diagrams of the rank 2 geometries obtained for Sz(8) which is the smallest case and which can still be completely determined using the help of MAGMA [8] or CAYLEY [7]. The whole classification of the geometries of Sz(8) is available in [14]. The classification of RWPRI geometries of rank > 2 for Sz(q), will be developed in a forthcoming paper. Finaly, in Section 9, we give some observations and also mention some future work that will be accomplished on similar problems.

Acknowledgement. We would like to thank Francis Buekenhout for many interesting discussions while solving this problem. We also want to thank Hendrik Van Maldeghem for explaining us the geometric meaning of some maximal subgroups of Sz(q). We acknowledge an argument simplifying the final result due to J.A.Thas. Finally, we also thank Michel Dehon for his comments on the preliminary version of this paper.

2. Definitions and notation

In a first paragraph, we recall the basic concepts of incidence geometry. For a more detailed discussion about incidence geometry, we refer to [2]. In a second paragraph, we give the construction of the Suzuki groups Sz(q) and some properties that will play an important role in this paper. For a more detailed introduction on Suzuki groups, see for example [16]. Finally, in a third paragraph, we fix some notation.

2.1. Groups and geometries

Most of the following ideas arise from [18] (see also [2], chapter 3). Let G be a group together with a finite family of subgroups (G_{i})_{i∈I}. We define the pregeometry Γ = Γ(G, (G_{i})_{i∈I}) as follows. The set X of elements of Γ consists of all cosets gG_{i}, g ∈ G, i ∈ I. We define an incidence relation * on X by :

g_{1}G_{i} * g_{2}G_{j} iff g_{1}G_{i} ∩ g_{2}G_{j} is non-empty in G.

The type function t on Γ is defined by t(gG_{i}) = i. The type of a subset Y of X is the set t(Y); its rank is the cardinality of t(Y) and we call |X| the rank of Γ. The Borel subgroup of the pre-geometry is the subgroup B = ∩_{i∈I}G_{i}. A flag is a set of pairwise incident elements of X and a chamber of Γ is a flag of type I. An element of type i is also called an i-element. The group G acts on Γ as an automorphism group, by left translation, preserving the type of each element. As in [10], we call Γ a geometry provided that every flag of Γ is contained in some chamber and we call Γ flag-transitive (FT) provided that G acts transitively on all chambers of Γ, hence also on all flags of any type J, where J is a subset of I. Assuming that Γ is a flag-transitive geometry and that F is a flag of Γ, the residue of F is the pre-geometry

Γ_{F} = Γ(∩_{j∈t(F)}G_{j}, (G_{i} ∩ (∩_{j∈t(F)}G_{j}))_{i∈I\t(F)})

and we readily see that Γ_{F} is a flag-transitive geometry. We call Γ firm (F) (resp. thick, thin) provided that every flag of rank |I| − 1 is contained in at least two (resp. three, exactly two) chambers. We call Γ residually connected (RC) provided that the incidence graph of each residue of rank ≥ 2 is a connected graph. We call Γ primitive (PRI) provided that G acts primitively on the set of i-elements of Γ, for each i ∈ I and we call Γ residually primitive (RPRI) if each residue Γ_{F} of a flag F is primitive for the group induced on Γ_{F} by the stabilizer G_{F} of F. We call Γ weakly primitive (WPRI) provided there exists some i ∈ I such that G acts primitively on the set of i-elements of Γ and we call Γ residually weakly primitive (RWPRI) provided that each residue Γ_{F} of a flag F is weakly primitive for the group induced on Γ_{F} by the stabilizer G_{F} of F. A subgroup H of a group G is called quasi-maximal if there exists a unique chain of intermediate subgroups from G to H. We call Γ quasi primitive (QPRI) provided that each subgroup G_{i} forming the geometry is a quasi-maximal subgroup of G. If Γ is a geometry of rank 2 with I = {0, 1} such that each of its 0-elements is incident with each of its 1-elements, we call Γ a generalized digon. We say that Γ satisfies the rank two intersection property (IP)_{2} if in every rank 2 residue of Γ other than a generalized digon, any two elements of the same type are incident with at most one element of the other type. We call Γ locally 2-transitive and we write (2T)_{1} for this, provided that the stabilizer G_{F} of any flag F of rank |I| − 1 acts 2-transitively on the residue Γ_{F}. The diagram of a firm, RC, FT 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 s_{i} which is |Γ_{F} | − 1, where F is any flag of type I\{i}, the number n_{i} of varieties of type i, which is the index of G_{i} in G, and the subgroup G_{i}. Elements i, j of I are not joined by an edge of the diagram provided that a residue Γ_{F} of type {i, j} is a generalized digon. Otherwise, i and j are joined by an edge endowed with three positive integers d_{ij}, g_{ij}, d_{ji} where g_{ij} (the gonality) is equal to half the girth of the incidence graph of a residue Γ_{F} of type {i, j} and d_{ij} (resp. d_{ji}), 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 Γ_{F}. On a picture of the diagram, this structure will often be depicted as follows.

[figure: d_{ij}; g_{ij}; d_{ji}; s_{i}; s_{j}; n_{i}; n_{j}; G_{i}; G_{j}]

If g_{ij} = d_{ij} = d_{ji} = n, then Γ_{F} is called a generalized n-gon and on a picture, we do not write d_{ij} and d_{ji}. As to notation for groups, we follow the conventions of the Atlas [9] up to slight variations. The symbol : stands for split extensions, the “hat” symbol ˆ.stands for non split extensions and the symbol × stands for direct products.

In this article, we will consider only geometries of rank 2, i.e. geometries Γ(G; G_{0}, G_{1}). We will call the quadruple (G, G_{0}, G_{1}, G_{01}) an RWPRI configuration (of rank 2) provided that G_{0} is maximal in G and G_{01} is maximal in both G_{0} and G_{1}.

Lemma 2.1. If Γ is an RWPRI configuration, then Γ is firm, residually connected and flagtransitive.

Proof. This is a direct consequence of the results obtained in [10]. □

This lemma implies that all geometries of rank 2 which are RWPRI are also F, RC, and FT. Thus the search of the rank 2 geometries of the Suzuki group Sz(q) that are F, RC, FT and RWPRI is reduced to the search of all RWPRI configurations.

The ordered pairs (Γ, G) and (Γ′, G) are isomorphic (resp. conjugate) if there exists an automorphism (resp. internal automorphism) of G mapping Γ onto Γ′.

2.2. The Suzuki simple groups Sz(q)

We recall the definition of the Suzuki groups as given in [16]. Let K be a field of characteristic 2 with |K| > 2. Let σ be an automorphism of K such that x^{σ^{2}} = x^{2} for each x in K. Let B be the 3-dimensional projective space over K and let (x_{0}, x_{1}, x_{2}, x_{3}) be the coordinates of a point of B. Let E be the plane defined by the equation x_{0} = 0 and put U = (0, 1, 0, 0)K. We introduce coordinates in the affine space B_{E} by x = \frac{x_{2}}{x_{0}} , y = \frac{x_{3}}{x_{0}} , and z = \frac{x_{1}}{x_{0}} . Finally, let D be the set of points of B consisting of U and all those points of B_{E} whose coordinates (x, y, z) satisfy the equation

z = xy + x^{σ+2} + y^{σ}

We denote by Sz(K, σ) the group of all projective collineations of B which leave D invariant.

Lemma 2.2. [16]

• If K is isomorphic to GF (2^{2e+1}) with e ≥ 1, then K admits exactly one automorphism σ with x^{σ^{2}} = x^{2} for all x in K.

• If K is isomorphic to GF (2^{2e}), then K does not possess an automorphism σ with x^{σ^{2}} = x^{2} for all x in K.

This theorem yields that we may write Sz(q) instead of Sz(K, σ) if K is isomorphic to GF (q) with q = 2^{2e+1}. The groups Sz(q) are the Suzuki groups named after Michio Suzuki who found them in 1960. The generalizations Sz(K, σ) are due to Rimhak Ree and Jacques Tits (see for example [19]). The set D is an ovoid, i.e. a non-empty point-set of a projective 3-space that satisfies the following three conditions.

1. No three points are collinear.

2. If p ∈ D, there exists a plane E of B with D ∩ E = {p}.

3. If p ∈ D and if E is a plane of B with D ∩ E = {p}, then all lines l through p which are not contained in E carry a point of D distinct from p.

For a, b ∈ K, we denote by τ(a, b) the mapping defined by

(x, y, z)^{τ(a,b)} = (x + a, y + b + a^{σ}x, z + ab + a^{σ+2} + b^{σ} + ay + a^{σ+1}x + bx)

We have τ(a, b)τ(c, d) = τ(a + c, ac^{σ} + b + d). For k ∈ K^{∗}, we define the collineation η(k) by

(x, y, z)^{η(k)} = (kx, k^{σ+1}y, k^{σ+2}z)

A straightforward computation shows that τ(a, b)η(k) = η(k)τ(ka, k^{σ+1}b). Let ω be the collineation of B defined by (x_{0}, x_{1}, x_{2}, x_{3})^{ω} = (x_{1}, x_{0}, x_{3}, x_{2}). The following lemma, due to Tits, will be used in a further section to study the structure of the stabilizer of a point of the ovoid.

Lemma 2.3. [19] Let K be a commutative field of characteristic 2 with |K| > 2 and assume that K admits an automorphism σ such that x^{σ^{2}} = x^{2} for all x ∈ K. If D is the point set defined above in the projective space of dimension 3 over K, then Sz(K, σ) acts doubly transitively on D. Moreover, if γ ∈ Sz(K, σ)_{U}, then there exists exactly one triple (k, a, b) ∈ K^{∗} × K × K with γ = η(k)τ(a, b), and if γ ∈ Sz(K, σ)\Sz(K, σ)_{U}, then there exists exactly one 5-tuple (k, a, b, c, d) ∈ K^{∗} × K × K × K × K with γ = η(k)τ(a, b)ωτ(c, d).

2.3. Some notation

There are four numbers that play an important role in the structure of the subgroups of Sz(q). They are respectively q, q − 1, q + r + 1 and q − r + 1 where r = \sqrt{2q}. We note q + r + 1 = α_{q} and q − r + 1 = β_{q}.

When n is a proper divisor of m, we write n |_{P} m. When n is a maximal divisor of m, we write n |_{M} m.

3. Some useful properties

In this section, we give some properties concerning q − 1, α_{q}, and β_{q}.

Lemma 3.1. (Cataldi, 1603) If 2^{2e+1} − 1 is a prime number, then 2e + 1 is a prime number.

The following lemma implies that if q − 1 is a prime number then the Suzuki group Sz(q) does not have any subgroup isomorphic to a Suzuki group of lower order. Let us remark that the converse is not true.

Lemma 3.2. If there exist s = 2^{2f+1} and an integer a such that q = s^{a}, then q − 1 is not a prime number.

Proof. s = 2^{2f+1} and q = 2^{2e+1} = 2^{a·(2f+1)} implies that 2e+1 is not a prime number. Suppose q − 1 is a prime number. Then Lemma 3.1 says that 2e + 1 must be a prime number. Thus we have a contradiction. □

Lemma 3.3. The numbers q − 1, α_{q} and β_{q} are pairwise coprime.

Proof. (q − 1, α_{q}) = p implies that p divides α_{q} − (q − 1) and α_{q} + (q − 1). Thus p divides 2^{e+1} + 1 and 2^{e+1} + 2. This implies p = 1. By a similar argument, we see that (q − 1, β_{q}) = p implies that p divides both 2^{e+1} − 2 and 2^{e+1} − 1. Thus p = 1. At last, suppose (α_{q}, β_{q}) = p ≠ 1. Then p divides α_{q} − β_{q} which is a power of 2. Thus p is a even number dividing two odd numbers, a contradiction. □

Lemma 3.4. Let s = 2^{2e+1} and q = s^{2f+1}. Then s^{2} + 1 divides q^{2} + 1.

Proof. s^{2} + 1 = 2^{4e+2} + 1 and q^{2} + 1 = 2^{(4e+2)(2f+1)} + 1. Thus we have

\frac{q^{2} + 1}{s^{2} + 1} = ∑_{i=1}^{2f+1}(−1)^{i+1} · 2^{(4e+2)(2f+1−i)} □

Lemma 3.5. Let s = 2^{2e+1} and q = s^{2f+1}. Then, s^{2} + 1 does not divide α_{q} nor β_{q}.

Proof. The following argument is due to J.A. Thas. Assume first that s^{2} + 1 | α_{q}. Then

2^{2(2e+1)} + 1 | 2^{(2e+1)(2f+1)} + 2^{2ef+e+f+1} + 1

Suppose first that f is odd. Then

\begin{aligned}2^{2(2e+1)} + 1 & | & (2^{2(2e+1)f} + 1 − 1)2^{2e+1} + 2^{2ef+e+f+1} + 1 \\ & | & −2^{2e+1} + 2^{2ef+e+f+1} + 1 \end{aligned}

Because f is odd, we can write f = 2r + 1 with r an integer. Then

\begin{aligned}2^{2(2e+1)} + 1 & | & −2^{2e+1} + 2^{4er+2e+2r+1+e+1} + 1 \\ & | & −2^{2e+1} + 2^{2(2e+1)r+3e+2} + 1 \end{aligned}

Suppose that r is odd. Then

2^{2(2e+1)} + 1 | −2^{2e+1} + (2^{2(2e+1)r} + 1 − 1)2^{3e+2} + 1
| −2^{2e+1} − 2^{3e+2} + 1
| 2^{2e+1} + 2^{3e+2} − 1

But 2^{4e+2} + 1 > 2^{2e+1} + 2^{3e+2} − 1, thus we have a contradiction. Suppose that r is even. Then r = t + 1, with t odd and we have the following.

2^{2(2e+1)} + 1 | −2^{2e+1} + (2^{2(2e+1)t} + 1 − 1)2^{7e+4} + 1
| −2^{2e+1} − 2^{7e+4} + 1
| (2^{2(2e+1} + 1 − 1)2^{3e+2} + 2^{2e+1} + 1
| 2^{3e+2} − 2^{2e+1} − 1

But 2^{4e+2} + 1 > 2^{3e+2} − 2^{2e+1} − 1, thus we have a contradiction. Hence we have proved that f cannot be odd. The same reasoning can be made for f even. And the same arguments can be used to show that 2^{2(2e+1)} + 1 cannot divide β_{q}. □

Lemma 3.6. Let s = 2^{2e+1} and q = s^{2f+1}, with e and f some positive integers. Then α_{q} and β_{q} are not prime.

Proof. Easy to see thanks to Lemmas 3.4, 3.5. □

Lemma 3.7. (Fermat, 1640) If p is a prime number then p divides 2^{p} − 2.

4. The maximal subgroups of Sz(q)

Table 1 gives a description of the conjugacy classes of maximal subgroups of Sz(q). Those results have been obtained by Suzuki [17]. It is from these subgroups that we start to classify all RWPRI configurations in the subgroup lattice of Sz(q). For a Suzuki simple group Sz(q), with q = 2^{2e+1}, if 2e + 1 is a prime number, there are no proper subgroups isomorphic to a Suzuki simple group. This implies that if 2e + 1 is a prime number, the group Sz(q) has only four conjugacy classes of maximal subgroups. Some Suzuki groups may have subgroups of Suzuki type which are not isomorphic. For example, Sz(32768) has maximal subgroups that are respectively isomorphic to Sz(8) and Sz(32). This means we don’t have five classes of maximal subgroups there but six. In fact, the number of non-isomorphic classes of Sz(s) is equal to the number of prime divisors of 2e + 1. Dividing 2e + 1 by each of them gives us a number 2f + 1 for which there exists a Sz(2^{2f+1}) which is a maximal subgroup of Sz(q).

5. The maximal subgroups of the maximal subgroups of Sz(q)

Lemma 5.1. The maximal subgroups of (qˆ.q) : (q − 1) have one of the following structures. • (qˆ.q) : n where n is a maximal divisor of q − 1.

• q : (q − 1)

Proof. The subgroup G_{0} is the stabilizer of a point. Let us take U as the fixed point. Then, a complete description of this group is given in [16]. Lüneburg also shows that the stabilizer of a circle is of order q(q − 1). A circle is obtained by intersecting the ovoid with a plane. If we take the plane Π defined by a cartesian equation x = 0, we see that the τ(0, b) and the η(k) stabilize this plane for b ∈ GF (q) and k ∈ GF (q)^{∗} = GF (q) − {0}. Looking at the subgroup generated by these collineations, we have a subgroup of order q(q − 1) which is the stabilizer of a circle. This group has a normal subgroup formed by the τ(0, b), of order q.

Structure  Order  Index Description
(qˆ.q) : (q − 1)  q^{2} · (q − 1)  q^{2} + 1 Normalizer of a 2-Sylow,

stabilizer of a point of the ovoid

D_{2(q−1)} 2 · (q-1) \frac{(q^{2}+1)·q^{2}}{2} Stabilizer of a pair of

points of the ovoid

α_{q} : 4 α_{q} · 4 \frac{q^{2}(q−1)}{4β_{q}} Normalizer of a

α_{q}-Sylow

β_{q} : 4 β_{q} · 4 \frac{q^{2}(q−1)}{4α_{q}} Normalizer of a

β_{q}-Sylow

Sz(2^{2f+1})  (s^{2} + 1) · s^{2} · (s − 1)
with 2f + 1 |_{M} 2e + 1

Table 1: The maximal subgroups of Sz(q)

And its structure is q : (q − 1). We denote it by G_{C}. We still have to show that it is a maximal subgroup of G_{0}. Thanks to Lemma 2.3, every element of G_{0} can be written in a unique form as η(k)τ(a, b) with a, b ∈ GF (q) and k ∈ GF (q)^{∗}. Thus we see that the only elements we could add are of the form τ(a, 0) with a ∈ GF (q), a ≠ 0. If we add any of these elements to G_{C}, the η(k) act such that every τ(a, 0) is generated, and thus G_{0} is generated. We can then conclude that G_{C} is maximal in G_{0}.

Taking all τ(a, b), we obtain a subgroup of structure qˆ.q which is normal in G_{0}. The subgroup formed by the η(k), with k ∈ GF (q)^{∗} is a cyclic group of order q − 1. It then has maximal subgroups which are the subgroups of order n with n a maximal divisor of q − 1. If we take one generator of such a subgroup, namely η(k) and we look at the subgroup generated by the τ(a, b) and this η(k), we obtain a subgroup of order q^{2} · n. Again, qˆ.q is normal in it and thus this subgroup has structure (qˆ.q) : n. The only elements that are missing use η(l) with l not appearing in the multiplicative group generated by k. This multiplicative group is a maximal subgroup of GF (q)^{∗}. We can conclude that (qˆ.q) : n is maximal in G_{0}.

The group generated by all the τ(a, 0) and the η(k), with a ∈ GF (q) and k ∈ GF (q)^{∗} is G_{0}. Thus no other maximal subgroups can be found. □

Lemma 5.2. The maximal subgroups of n : 4 have the following structure.

• m : 4 where m is a maximal divisor of n

• n : 2

Proof. Obviously, n : 4 has a maximal subgroup whose structure is n : 2. In n : 4, there exists a normal subgroup isomorphic to m for each divisor m of n. Extending this cyclic group by 4, we obtain a group of order 4m which has structure m : 4 because (m, 4) = 1. If m is a maximal divisor of n, then [n:4 : m:4] is a prime number, hence m:4 is a maximal subgroup of n:4. □

Lemma 5.3. The maximal subgroups of D_{2(q−1)} have the following structure.

• the cyclic group of order q − 1

• D_{2n} where n is a maximal divisor of q − 1. If q − 1 is a prime number then D_{2} is the cyclic group of order 2.

Proof. Straightforward □

As to the maximal subgroups that are isomorphic to some Suzuki simple group, we refer to Table 1.

6. Rank 2 RWPRI geometries

Let G be a group. Let G_{0} and G_{1} be subgroups of G and let G_{01} = G_{0} ∩ G_{1}. The RWPRI condition in rank 2 asks that either G_{0} or G_{1} must be a maximal subgroup of G and that G_{01} must be a maximal subgroup of G_{0} and G_{1}. We use the following method to determine all the RWPRI geometries of the Suzuki group Sz(q). We choose a subgroup G_{0} and we impose that it must be a maximal subgroup of Sz(q). Then we determine all the G_{01} possible groups. They are the maximal subgroups of G_{0}. Finally, we look at all the G_{1} candidates which must contain G_{01} as a maximal subgroup and which must be different from G_{0}.

Lemma 6.1. Let G ≅ Sz(q). There are no RWPRI geometries of rank > 1 with G_{0} ≅ (qˆ.q) : (q − 1).

Proof. We start from the maximal subgroups of G_{0}. They have been determined in Lemma 5.1. We see that the only subgroups of Sz(q) that contain one of the G_{01} candidates are thus isomorphic to G_{0}. Because the normalizer of each G_{01} candidate must be a subgroup of G_{0}, we obtain that G_{1} must be equal to G_{0} which is a contradiction to the fact that G_{01} is their intersection. □

Lemma 6.2. Let e > 0 be a natural number. Let q = 2^{2e+1} and let G ≅ Sz(q) If 2e + 1 and q − 1 are prime numbers, the RWPRI rank 2 geometries of G with G_{0} ≅ D_{2(q−1)} are isomorphic to one of the geometries appearing in Table 2. If 2e + 1 is prime and q − 1 is not prime, the only RWPRI rank 2 geometry of G with G_{0} ≅ D_{2(q−1)} is obtained with G_{1} ≅ q : (q-1). If 2e + 1 is not prime, the RWPRI rank 2 geometries of G with G_{0} ≅ D_{2(q−1)} are isomorphic to one of the geometries appearing in Table 3.

Proof. By Lemma 5.3, G_{01} can be isomorphic to either q − 1 of D_{2n} where n is a maximal divisor of q − 1.

Suppose first that G_{01} = q − 1. The G_{1} candidates are neither subgroups of α_{q} : 4, nor of β_{q} : 4 because of Lemma 3.3. The normalizer of a q − 1 is G_{0}. Thus two dihedral groups D_{2(q−1)} never intersect on a cyclic group of order q − 1 in Sz(q), and G_{1} cannot be a D_{2(q−1)}. So the only candidates must be subgroups of (qˆ.q) : (q − 1) or Sz(s) with q = s^{a} for some integer a.

Suppose G_{1} < (qˆ.q) : (q − 1). Then, in order to have G_{01} maximal in G_{1} the only candidate G_{1} is q : (q − 1). This gives rise to an RWPRI configuration. If we fix G_{1} and

G_{1}  G_{01} #Γ
q : (q − 1)  q − 1 1
D_{2(q−1)} 2 \frac{(q−2)(q+4)}{4(2e+1)} + 1
D_{2n} where n is a prime divisor of α_{q} or β_{q} 2  \frac{(q−2)(q+2)}{4(2e+1)} + 1
2^{2}  2  \frac{q−2}{2(2e+1)}
4  2  \frac{q−2}{2(2e+1)} + 1
Table 2: G_{0} = D_{2(q−1)}, and 2e + 1 and q − 1 are prime numbers
G_{1}  G_{01}  #Γ
q : (q − 1)  q − 1 1
Sz(s) iff s − 1 is a maximal divisor of q − 1 D_{2(s−1)} 1
Table 3: G_{0} = D_{2(q−1)} and 2e + 1 is not prime

G_{01} (this can be done because all subgroups isomorphic to G_{01} are conjugate in G_{1}), there is only one D_{2(q−1)} intersecting G_{1} in G_{01}. Thus we have only one such configuration up to isomorphism.

Suppose G_{1} < Sz(s). As Sz(s) exists, it implies that q − 1 is not a prime number. No maximal subgroup of Sz(s) contains a cyclic group of order q − 1. Thus G_{1} must be Sz(s). But q − 1 is not a maximal subgroup of Sz(s).

Now suppose that G_{01} = D_{2n} where n is a maximal divisor of q − 1. Let us first examine the case where q − 1 is a prime number. Then G_{01} is a cyclic group of order 2. This gives rise to the following candidates for G_{1}.

• D_{2(q−1)}

• D_{2m} where m is a prime divisor of α_{q} or β_{q}.

• 2^{2}

• 4

All of these subgroups can be taken as G_{1} to form an RWPRI configuration.

Suppose G_{1} = D_{2(q−1)}. We can fix G_{0} and G_{01} because all subgroups isomorphic to G_{01} are conjugate in G_{0}. We know that there are \frac{q^{2}·(q^{2}+1)}{2} subgroups isomorphic to G_{0} in Sz(q). We also know that G_{0} contains q − 1 subgroups of order 2. And the number of involutions in Sz(q) is equal to (q^{2}+1)(q−1). Thus we can conclude that there are \frac{q^{2}}{2} subgroups isomorphic to G_{0} intersecting in the same G_{01}. If we want to fix the types of our geometries, G_{01} cannot transform one of these geometries into another. If we do not fix the types, then we have to count how many of our geometries are self-dual by conjugacy. Suppose we have a geometry Γ(Sz(q); D_{2(q−1)} = G_{0}, D_{2(q−1)} = G_{1}). We know that the normalizer of G_{01} is a group of order q^{2} isomorphic to qˆ.q. By looking at the action of H = qˆ.q/G_{01} on the \frac{q^{2}}{2} subgroups isomorphic to D_{2(q−1)} containing G_{01}, we see that it is transitive and hence strictly transitive because its order is equal to its degree. Because H contains q − 1 involutions, there are q − 1 geometries that are self-dual, and the \frac{q^{2}−2q}{2} elements of order 4 spread the \frac{q^{2}}{2} subgroups into squares. Every square appears two times, that is one time in each way. And no two disjoint squares can have a common vertex. This means that through every of the \frac{q^{2}}{2} points, there are \frac{q^{2}−2q}{4} disjoint squares that fuse \frac{q^{2}−2q}{2} geometries inside sets of 2 geometries. So finally, we have q − 1 + \frac{q^{2}−2q}{4} = 1 + \frac{(q−2)(q+4)}{4} non-conjugate geometries in this case. In Aut(Sz(q)), the stabilizer of G_{0} and G_{01} is a cyclic group of order 2(2e+1). Its normalizer is two times bigger. The normalizer of G_{1} is G_{1} : (2e+1). And each subgroup isomorphic to G_{1} : (2e+1) contains q −1 subgroups isomorphic to Z_{2(2e+1)}. This implies that there are two subgroups isomorphic to G_{1} : (2e + 1) containing the same Z_{2(2e+1)}. Thus one geometry is fixed under the action of Aut(Sz(q)) and all the others are fused inside sets of 2e + 1. That is, we have 1 + \frac{(q−2)(q+4)}{4(2e+1)} non-isomorphic RWPRI configurations and we observe that this number is indeed an integer thanks to Lemma 3.7.

Suppose G_{1} = D_{2m} where m is a prime divisor of α_{q} or β_{q}. We can decide to fix it without loss of generality. All its subgroups isomorphic to G_{01} are conjugate in it. This implies that we can also fix G_{01}. The normalizer of G_{1} in G is m:4. So the stabilizer of G_{1} and G_{01} is a cyclic group of order 4. And there are \frac{q^{2}}{2} subgroups isomorphic to G_{0} that intersect G_{1} in G_{01}. The cyclic group of order 4 fuses them inside sets of two. So there are \frac{q^{2}}{4} non-conjugate geometries. The subgroup which stabilizes G_{0} and G_{01} in Aut(G) is isomorphic to Z_{2(2e+1)}. The subgroup stabilizing G_{1} in Aut(G) is isomorphic to G_{1} : (2e + 1). Again, a simple counting argument permits us to conclude that there is only one subgroup isomorphic to G_{1} which is fixed under the action of Z_{2(2e+1)}. Thus, from the \frac{q^{2}}{4} non-conjugate geometries, one is fixed under the action of Aut(G) and all other are fused inside sets of 2e + 1 geometries. This means we have 1 + \frac{(q−2)(q+2)}{4(2e+1)} non-isomorphic RWPRI configurations of that kind. And, again, thanks to Lemma 3.7, we observe that this number is indeed an integer.

Suppose G_{1} = 2^{2}. Then G_{1} contains three subgroups of order 2. Also, G_{1} is contained in the stabilizer of some point. In each point-stabilizer, there are q−1 Z_{2}. We must combine two of them to obtain subgroups isomorphic to 2^{2}. We first choose one Z_{2}. Then, the second one can be chosen in \frac{q−2}{2} ways. But there we have taken each possibility 3 times. So the number of 2^{2} is \frac{(q^{2}+1)(q−1)(q−2)}{6}. From this, we can count the number of 2^{2} containing the same Z_{2}. There are \frac{q−2}{2} such subgroups. Thus, by fixing G_{0} and G_{01}, we have \frac{q−2}{2} possible G_{1}. These give \frac{q−2}{2} non-conjugate geometries. The normalizer of G_{1} in Aut(G) is G_{1} : (2e + 1) and this subgroup does not contain the stabilizer of G_{0} and G_{01} in Aut(G). Thus no configuration of that kind is fixed under the action of Aut(G). This means that all of them are fused in sets of 2e + 1 geometries. Hence we have \frac{q−2}{2(2e+1)} non-isomorphic RWPRI configurations like that. Let us observe that in the smallest case (i.e. for q = 8), there is only one geometry of that kind up to isomorphism.

Suppose G_{1} = 4. Let us count the number of cyclic groups of order 4 contained in a Sz(q). Each of them is contained in the stabilizer of a point (there are q^{2} + 1 such stabilizers). In each stabilizer, the τ(a, b) with a ≠ 0 are elements of order 4. There are (q − 1)q such elements. In a cyclic group of order 4, there are 2 elements of order 4. Thus the number of cyclic groups of order 4 in Sz(q) equals (q^{2} + 1)(q − 1)\frac{q}{2}. We already know that the number of Z_{2} in Sz(q) is (q^{2} + 1)(q − 1). And it is well known that every Z_{4} contains only one Z_{2}. Hence the number of Z_{4} containing the same Z_{2} is \frac{q}{2}. Now we fix G_{0} and we can also fix G_{01} because all the subgroups that are isomorphic to G_{01} are conjugate in G_{0}. The stabilizer of

G_{1}  G_{01} #Γ
α_{q} : 4 4 \frac{q−2)}{2(2e+1)}
n : 4 where n is a prime divisor of β_{q} 4  \frac{q−2}{2(2e+1)} + 1
4 × 2  4  \frac{q−2}{2(2e+1)}
Table 4: The case G_{0} = α_{q} : 4 and α_{q} prime.
G_{1}  G_{01}  #Γ
Sz(s) iff α_{s} or β_{s} is a maximal divisor of α_{q} α_{s} : 4 or β_{s} : 4 1
Table 5: The case G_{0} = α_{q} : 4 and α_{q} not prime.

G_{0} and G_{01} is G_{01}. Thus it fixes G_{1} also. This means we have \frac{q}{2} non-conjugate geometries. The normalizer of G_{1} in Aut(G) being a subgroup isomorphic to 4 : (2e + 1), it contains the stabilizer of G_{0} and G_{01} in Aut(G). This means that there is one configuration which is fixed under the action of Aut(G) and all the others are fused under Aut(G) inside sets of 2e + 1 geometries. Thus we have 1 + \frac{q−2}{2(2e+1)} non-isomorphic RWPRI configurations of that kind.

Suppose now that q − 1 is not a prime number. Then G_{01} is a dihedral group D_{2n}. This means that the G_{1} candidates will neither be subgroups of α_{q} : 4, nor of β_{q} : 4. Here we have two candidates for G_{1}. They are respectively D_{2(q−1)} and Sz(s) if s − 1 = n. The case G_{1} = D_{2(q−1)} is not possible. This is due to the fact that N_{Sz(q)}(D_{2n}) = D_{2n}. While fixing G_{0}, we can also fix G_{01} because all the subgroups isomorphic to G_{01} in G_{0} are conjugate. It is easy to see that there is only one subgroup isomorphic to Sz(s) which contains G_{01}. So the G_{1} is uniquely determined. This gives us only one geometry up to conjugacy and thus up to isomorphism. □ An example of a configuration with G_{1} = Sz(s) can be obtained in Sz(512) by taking as G_{0} (resp. G_{1}, G_{01}) a subgroup isomorphic to D_{2(511)} (resp. Sz(8), D_{14}).

Lemma 6.3. Let e > 0 be a natural number. Let q = 2^{2e+1} and let G ≅ Sz(q) If α_{q} is a prime number, the RWPRI rank 2 geometries of G with G_{0} ≅ α_{q} : 4 are isomorphic to one of the geometries appearing in Table 4. If α_{q} is not a prime number, the RWPRI rank 2 geometries of G with G_{0} ≅ α_{q} : 4 are isomorphic to one of the geometries appearing in Table 5.

Proof. By Lemma 5.2, G_{01} can be isomorphic to either α_{q} : 2 or n : 4 where n is a maximal divisor of α_{q}.

Suppose first that G_{01} = n : 4. Let us first consider the case where n = 1 (i.e. α_{q} is a prime number). In this case, several candidates appear for G_{1}.

• 4 × 2

• m : 4 where m | β_{q}, and m is a prime number.

• α_{q} : 4

All these cases give rise to RWPRI candidates.

Suppose G_{1} = 4×2. Let us count the number of subgroups isomorphic to 4×2 in Sz(q). Each of them is contained in the stabilizer of a point. In each point-stabilizer, there are (q − 1)\frac{q}{2} cyclic groups of order 4, as we saw before. The 4×2 are constructed by adding to these Z_{4} an element of order 2. This element must be different from the one in the Z_{4}. Thus we have q−2 such possible elements. But a 4 × 2 contains 3 subgroups of order 2. This implies that there are \frac{q−2}{2} subgroups isomorphic to 4 × 2 which contain the same Z_{4}. And every 4 × 2 contains 2 subgroups of order 4. Thus the number of 4×2 is (q^{2} +1)(q −1)\frac{q}{2} · \frac{q−2}{2} · \frac{1}{2} = \frac{(q^{2}+1)q(q−1)(q−2)}{8}. Now let us count the number of non-conjugate and non-isomorphic geometries. We can fix G_{0}. All the subgroups isomorphic to Z_{4} are conjugate in G_{0}. This implies that we can also fix G_{01}. The stabilizer of G_{0} and G_{01} being G_{01}, we have \frac{q−2}{2} non-conjugate geometries. The (2e + 1) classes of subgroups isomorphic to 4 × 2 in G are fused under the action of Aut(G). This means that the normalizer of a subgroup isomorphic to 4 × 2 will not contain elements of order (2e + 1). The stabilizer of G_{0} and G_{01} in Aut(G) is a subgroup isomorphic to 4 : (2e + 1). Thus, there is no configuration which is fixed under the action of Aut(G). Then, all the non-conjugate configurations are fused under the action of Aut(G) inside sets of 2e + 1 geometries. Hence we have \frac{q−2}{2(2e+1)} non-isomorphic RWPRI configurations like that. Let us observe that in the smallest case (i.e. for q = 8), there is only one such geometry up to isomorphism.

Suppose G_{1} = m:4. The subgroup G_{1} has its normalizer in β_{q} : 4. Thus it is selfnormalizing. This gives us the number of subgroups isomorphic to m:4 in Sz(q). There are \frac{(q^{2}+1)q^{2}(q−1)}{4m} such subgroups. Each of them contains m cyclic subgroups of order 4. And we already know that there are \frac{(q^{2}+1)q(q−1)}{2} cyclic groups of order 4 in Sz(q). Thus every Z_{4} is contained in \frac{q}{2} subgroups m:4. We can fix G_{0}. All the subgroups isomorphic to Z_{4} are conjugate in G_{0}. Thus we can also fix G_{01}. The stabilizer of G_{0} and G_{01} is G_{01}. This implies that there are \frac{q}{2} non-conjugate geometries here. The stabilizer of G_{1} and G_{01} in Aut(G) being isomorphic to 4 : (2e + 1), there is only one configuration which is fixed under the action of Aut(G) and all the others are fused inside sets of 2e + 1 geometries. Thus we have 1 + \frac{q−2}{2(2e+1)} non-isomorphic RWPRI configurations in that case.

Suppose G_{1} = α_{q} : 4. Then there are \frac{q}{2} subgroups isomorphic to G_{1} that contain the same Z_{4}. One of them is G_{0} so we have here \frac{q}{2} − 1 non-conjugate geometries if we decide to fix the types. It is easy to see that N_{G}(Z_{4}) is a group which is of order 2q and which is isomorphic to qˆ.2. By looking at the action of (qˆ.2)/4 on the set of \frac{q}{2} subgroups isomorphic to α_{q} : 4, we see that this group is transitive and hence strictly transitive. It is also easy to see that all its elements (except the identity) are involutions. This means that all the \frac{q}{2} − 1 geometries we found are self-dual. Thus there are \frac{q−2}{2} non-conjugate geometries. The stabilizer of G_{0} and G_{01} in Aut(G) is isomorphic to 4 : (2e + 1). Thus there is no geometry fixed under the action of Aut(G) and all of them are fused inside sets of 2e+1 elements. This means we have \frac{q−2}{2(2e+1)} non-isomorphic RWPRI configurations of that kind. Let us observe that in the smallest case (i.e. for q = 8), there is only one geometry up to isomorphism.

Now suppose n ≠ 1, which means that α_{q} is not a prime number. Then G_{1} is not a subgroup of β_{q} : 4 (resp. D_{2(q−1)}) because (α_{q}, β_{q}) (resp. (α_{q}, q − 1)) = 1. Two candidates arise for G_{1}. They are α_{q} : 4 and Sz(s) if and only if α_{s} or β_{s} is a maximal divisor of α_{q}.

The normalizer N_{Sz(q)}(n : 4) is n : 4. Thus the case G_{1} = α_{q} : 4 leads to a contradiction.

G_{1}  G_{01} #Γ
β_{q} : 4 4 \frac{q−2)}{2(2e+1)}
n : 4 where n is a prime divisor of β_{q} 4  \frac{q−2}{2(2e+1)} + 1
4 × 2  4  \frac{q−2}{2(2e+1)}
Table 6: The case G_{0} = β_{q} : 4 and β_{q} prime.
G_{1}  G_{01}  #Γ
Sz(s) iff α_{s} or β_{s} is a maximal divisor of β_{q} α_{s} : 4 or β_{s} : 4 1
Table 7: The case G_{0} = β_{q} : 4 and β_{q} not prime.

We then have just one candidate left, that is the one with G_{1} = Sz(s). We can fix G_{0} and G_{01} because G_{0} conjugates all its subgroups isomorphic to G_{01}, the stabilizer being G_{01}. And there is only one Sz(s) containing G_{01}. Thus there is only one such geometry up to conjugacy, and hence up to isomorphism.

Suppose that G_{01} = α_{q} : 2. This implies that G_{1} must be a subgroup of Sz(s). So q − 1 cannot be a prime number. The order of G_{01} must divide the order of G_{1}. Suppose G_{1} is a subgroup of (sˆ.s) : (s − 1) or D_{2(s−1)}. Then α_{q} divides s − 1 which is a contradiction to the fact that (q − 1, α_{q}) = 1 and s − 1 | q − 1. Suppose G_{1} is a subgroup of α_{s}:4 (resp. β_{s} : 4). Then the order of G_{1} is smaller than the order of G_{01} which is also a contradiction. Finally, suppose G_{1} = Sz(s). Then G_{01} must be a maximal subgroup of Sz(s). This yields s − 1 = α_{q} and s − 1 | q − 1, a contradiction. □

Lemma 6.4. Let e > 0 be a natural number. Let q = 2^{2e+1} and let G ≅ z(q). If β_{q} is a prime number, the RWPRI rank 2 geometries of G with G_{0} ≅ β_{q} : 4 are isomorphic to one of the geometries appearing in Table 6. If β_{q} is not a prime number, the only RWPRI rank 2 geometries of G with G_{0} ≅ β_{q} : 4 are isomorphic to one of the geometries appearing in Table 7.

Proof. Same as proof of lemma 6.3 □

At this stage, we have determined the RWPRI configurations for the rank 2 geometries of the Suzuki groups Sz(q) where 2e + 1 is a prime number. If 2e + 1 is not a prime number, Sz(q) may have other maximal subgroups which are isomorphic to Sz(s). Before starting the discussion about this case, we demonstrate a few lemmas that are used in the lemma giving the RWPRI rank 2 geometries of Sz(q) with G_{0} ≅ Sz(s).

Lemma 6.5. For every subgroup isomorphic to (sˆ.s) : (s − 1), there is only one subgroup isomorphic to Sz(s) containing it.

Proof. Let P be a subgroup isomorphic to (sˆ.s) : (s − 1). Every subgroups isomorphic to Sz(s) are conjugate in Sz(q). We only have to show that N_{Sz(q)}(P) ≤ Sz(s) in order to prove that there exists only one Sz(s) containing P. Sz(s) has an orbit of length s^{2} + 1. In this orbit, P fixes a point x and exchanges all the s^{2} other points. We know that N_{Sz(q)}(P) fixes x. It is thus a subgroup of the stabilizer G_{x} of x which has structure (qˆ.q) : (q − 1). In G_{x}, there is a cyclic subgroup Z_{q−1} which has exactly two fixed points. And so does Z_{s−1} ≤ Z_{q−1}. Thus the second fixed point is in the orbit of s^{2} points. We also know that P has one orbit of length 1, one of length s^{2} and all the others of length ≥ s^{2}. It can be easily seen that the other orbits have length s^{2}(s − 1). Because s ≠ 2, the normalizer N_{Sz(q)}(P) must stabilize the orbit of 1 point and the orbit of s^{2} points. Thus it must be a subgroup of Sz(s). □

Lemma 6.6. For every subgroup isomorphic to D_{2(s−1)}, there is only one subgroup isomorphic to Sz(s) containing it.

Proof. The normalizer N_{Sz(q)}(D_{2(s−1)}) must be a subgroup of D_{2(q−1)}. Because q −1 and s−1 are odd numbers, D_{2(s−1)} is self-normalizing in D_{2(q−1)}. □

Lemma 6.7. [17] Let Π be the partition consisting of four classes of conjugate subgroups of Sz(q):

• the Sylow 2-subgroups of Sz(q),

• one class of cyclic groups of order q − 1,

• one class of cyclic groups of order q + r + 1,

• one class of cyclic groups of order q − r + 1. If W is a subgroup of Sz(q), then there exists X ∈ Π with W ⊆ N_{Sz(q)}(X) or W is isomorphic to Sz(s) with s ≤ q.

Lemma 6.8. For every subgroup isomorphic to α_{s} : 4, there is only one subgroup isomorphic to Sz(s) that contains it.

Proof. The number of Sz(s) in Sz(q) is equal to [Sz(q) : Sz(s)] because N_{Sz(q)}(Sz(s)) = Sz(s). By lemma 6.7, the normalizer N_{Sz(q)}(α_{s}:4) is either Sz(s) or a subgroup of α_{q} : 4 or β_{q} : 4. Suppose it is Sz(s). Then Sz(s) is not simple, a contradiction. So it must be a subgroup of either α_{q} : 4 or β_{q} : 4. Anyway, α_{s} : 4 is self-normalizing in the one that contains it. Thus N_{Sz(q)}(α_{s} : 4) = α_{s} : 4 and the number of α_{s} : 4 in Sz(q) is equal to [Sz(q) : α_{s} : 4]. Every Sz(s) contains exactly [Sz(s) : α_{s} : 4] subgroups isomorphic to α_{s}:4. Thus the number of Sz(s) containing the same α_{s} : 4 is equal to 1. □

Lemma 6.9. Let e>0 be a natural number. Let q =2^{2e+1} and let G≅Sz(q). Let G_{0} ≅Sz(s) be a maximal subgroup of Sz(q) and suppose G_{1} is not one of D_{2(q−1)}, α_{q} : 4, or β_{q} : 4 (these cases have been analysed in Lemmas 6.2, 6.3 and 6.4. Then the RWPRI rank 2 geometries of G with G_{0} ≅ Sz(s) are isomorphic to one of the geometries appearing in Table 8.

G_{1}  G_{01}  #Γ
(qˆ.s) : (s − 1)  (sˆ.s) : (s − 1) 1
Sz(s′) when it is a maximal subgroup and s′ ≠ s Sz(t)  1
D_{2n} where s − 1 |_{M} n |_{P} q-1  D_{2(s−1)}  1
n : 4 where α_{s} |_{M} n |_{P} α_{q} or β_{q}  α_{s} : 4  1
n : 4 where β_{s} |_{M} n |_{P} α_{q} or β_{q}  β_{s} : 4  1
Table 8: The case G_{0} = Sz(s).

Proof. Recall that the maximal subgroups of Sz(s) have one of the following structures. • (sˆ.s) : (s − 1)

• D_{2(s−1)}

• α_{s}:4

• β_{s}:4

• Sz(t) if and only if there exists a natural number b such that s = t^{b}.

Suppose G_{01} = (sˆ.s) : (s − 1). Because G_{01} is the stabilizer of a point of the subovoid, G_{1} must be either a subgroup of Sz(s) or (qˆ.q) : (q − 1). Lemma 6.5 implies that G_{1} cannot be a subgroup isomorphic to Sz(s). Thus we have to look at (qˆ.q) : (q − 1) and its subgroups to find some G_{1} candidate. All the elements of G_{01} are of the form η(k)τ(a, b) where a, b ∈ GF (s) and k ∈ GF (s)^{∗}. The subgroup ⟨G_{01}, τ(0, c)⟩ with c ∈ GF (q) − GF (s) has order q · s · (s − 1) because there are elements of the form η(z)τ(x, y) with x ∈ GF (s), y ∈ GF (q) and z ∈ GF (s)^{∗}. It contains G_{01} as a maximal subgroup because GF (s) is a maximal subfield of GF (q). It is easy to see that it has structure (qˆ.s) : (s − 1). The subgroup ⟨G_{01}, τ(c, 0)⟩ with c ∈ GF (q) − GF (s) has order q^{2}(s − 1) because it generates all elements of the form η(z)τ(x, y) with x, y ∈ GF (q), z ∈ GF (s)^{∗}. It contains (qˆ.s) : (s−1) and thus G_{01} is not maximal in it. The subgroup ⟨G_{01}, η(l)⟩ with l ∈ GF (q) − GF (s) is the group (qˆ.q) : (q − 1) which does not contain G_{01} as a maximal subgroup. We can fix G_{1} and G_{01} because G_{1} conjugates all its subgroups isomorphic to G_{01}. Because G_{01} is self-normalizing as it was demonstrated in the previous lemma, the stabilizer of G_{1} and G_{01} is G_{01}. And there is only one subgroup isomorphic to Sz(s) which contains G_{01}. This means we have here only one geometry up to conjugacy, and hence up to isomorphism.

Suppose G_{01} = D_{2(s−1)}. Because s − 1 divides q − 1 the only G_{1} candidates are Sz(s) and D_{2n} where n is a divisor of q − 1, and s − 1 is a maximal divisor of n. In fact, we can suppose that n is a proper divisor of q − 1 because the case n = q − 1 has been treated while determining the configurations with G_{0} = D_{2(q−1)}. Lemma 6.6 shows that two subgroups isomorphic to Sz(s) never intersect in a D_{2(s−1)}. Thus the only RWPRI candidates here are obtained by taking Sz(s) as G_{0}, and D_{2n} as G_{1} where n is a proper divisor of q − 1, and s − 1 is a maximal divisor of n. It is easy to see that there is only one such geometry up to conjugacy and hence up to isomorphism, for each such n.

Suppose G_{01} = α_{s} : 4. This implies that G_{1} will certainly not be a subgroup of D_{2(q−1)} or (qˆ.q) : (q − 1). The candidates are then another Sz(s) or n : 4 where n is a proper divisor either of α_{q} or β_{q} and α_{s} is a maximal divisor of n. Lemma 6.8 implies that G_{1} must be an n : 4. We can fix G_{1} because all subgroups isomorphic to G_{1} in Sz(q) are conjugate. We can also fix G_{01} because all the subgroups isomorphic to G_{01} are conjugate in G_{1}. The stabilizer of G_{1} and G_{01} is then G_{01}. And there is only one Sz(s) containing G_{01}. Thus there is only one such geometry up to conjugacy and hence up to isomorphism.

Suppose G_{01} = β_{s} : 4. Almost the same discussion as in the case G_{01} = α_{s} : 4 leads us to the conclusion that G_{1} must be n : 4 where n is a proper divisor either of α_{q} or of β_{q}, and β_{s} is a maximal divisor of n, and that there is only one such geometry up to conjugacy and hence up to isomorphism.

Finally, suppose G_{01} = Sz(t). The only candidates are subgroups isomorphic to Sz(s′) because of the order of Sz(t). The case s = s′ leads to a contradiction because, as Sz(q) is a simple group, and as the only subgroup between Sz(q) and Sz(t) is Sz(s) which is also simple, the normalizer of Sz(t) in Sz(q) is Sz(t). Thus we have here G_{1} = Sz(s′) with s ≠ s′ that gives a geometry. We can fix G_{0} and G_{01} here and there is only one subgroup isomorphic to G_{1} that contains G_{01}. Thus there is only one such geometry up to conjugacy and hence up to isomorphism. □

7. Classification Theorem

The results of the preceding section permit us to give a complete classification theorem giving all the firm, residually connected , flag-transitive and residually weakly primitive geometries of rank 2 for a Suzuki group Sz(q) up to isomorphism. The symbol #Γ in the tables below is the number of non-isomorphic geometries that arise from an RWPRI configuration. The last column of the tables tells whether or not the corresponding geometry satisfies (2T)_{1}. This condition is so easy to test in rank 2 that we do not give details about the results obtained.

Theorem 7.1. Let e > 0 be a natural number. Let q = 2^{2e+1} and let Sz(q) be the Suzuki simple group over GF(q). If the numbers 2e + 1 and q − 1 are prime, the F, RC, FT, RWPRI rank 2 geometries of Sz(q) are isomorphic to one of the geometries appearing in Table 9. If 2e + 1 is prime and q − 1 is not prime, the F, RC, FT, RWPRI rank 2 geometries of Sz(q) are isomorphic to one of the geometries appearing in Table 10. If 2e + 1 is not a prime number, the F, RC, FT, RWPRI rank 2 geometries of Sz(q) are isomorphic to one of the geometries appearing in Table 11.

Proof. This theorem is a compilation of the results obtained in Lemmas 6.1, 6.2, 6.3, 6.4, and 6.9. Thanks to Lemma 3.6, in the case 2e + 1 is not a prime number, we are sure that α_{q} and β_{q} are not prime numbers. □

G_{0}  G_{1}  G_{01} #Γ  (2T)_{1}
D_{2(q−1)} D_{2(q−1)}  2  \frac{(q−2)(q+4)}{4(2e+1)} + 1 NO
D_{2n} where n is a prime divisor of α_{q} or β_{q} 2  \frac{(q−2)(q+2)}{4(2e+1)} + 1 NO
2^{2}  2  \frac{q−2}{2(2e+1)}  NO
4  2  \frac{q−2}{2(2e+1)} + 1  NO
q : (q − 1)  q−1 1  YES

\underline{α_{q} prime}

α_{q} : 4  α_{q} : 4  4  \frac{q−2}{2(2e+1)}  NO
n : 4 where n is a prime divisor of β_{q}  4  \frac{q−2}{2(2e+1)} + 1  NO
4 × 2  4  \frac{q−2}{2(2e+1)}  NO

\underline{β_{q} prime}

β_{q} : 4  β_{q} : 4  4  \frac{q−2}{2(2e+1)}  YES iff q=8
n : 4 where n is a prime divisor of α_{q}  4  \frac{q−2}{2(2e+1)} + 1  NO
4 × 2  4  \frac{q−2}{2(2e+1)}  YES iff q=8

Table 9: The case 2e + 1 prime and q − 1 prime

G_{0}  G_{1}  G_{01} #Γ  (2T)_{1}
D_{2(q−1)} q : (q − 1)  q-1 1  YES

\underline{α_{q} prime}

α_{q} : 4 α_{q} : 4 4 \frac{q−2}{2(2e+1)} NO
n : 4 where n is a prime divisor of β_{q} 4  \frac{q−2}{2(2e+1)} + 1 NO
4 × 2 4 \frac{q−2}{2(2e+1)} NO
\underline{β_{q} prime}  NO
β_{q} : 4 β_{q} : 4 4 \frac{q−2}{2(2e+1)} NO
n : 4 where n is a prime divisor of α_{q} 4  \frac{q−2}{2(2e+1)} + 1 NO
4 × 2  4  \frac{q−2}{2(2e+1)}  NO

Table 10: The case 2e + 1 prime and q − 1 not prime

G_{0}  G_{1}  G_{01}  #Γ (2T)_{1}
D_{2(q−1)} q : (q − 1)  q − 1  1 YES
Sz(s) if s − 1 is a maximal divisor of q − 1  D_{2(s−1)}  1 NO
α_{q} : 4 Sz(s) if α_{s} or β_{s} is a maximal divisor of α_{q}  α_{s} : 4 or β_{s} : 4 1 NO
β_{q} : 4 Sz(s) if α_{s} or β_{s} is a maximal divisor of β_{q}  α_{s} : 4 or β_{s} : 4 1 NO
Sz(s) (qˆ.s) : (s − 1)  (sˆ.s) : (s − 1) 1 NO
Sz(s′) when it is a maximal subgroup and s′ ≠ s Sz(t)  1 NO
D_{2n} where s − 1 |_{M} n |_{P} q − 1  D_{2(s−1)}  1 NO
n : 4 where α_{s} |_{M} n |_{P} α_{q} or β_{q}  α_{s} : 4  1 NO
n : 4 where β_{s} |_{M} n |_{P} α_{q} or β_{q}  β_{s} : 4  1 NO

Table 11: The case 2e + 1 not prime (and thus q − 1 not prime)

\begin{matrix}e & #(q − 1) & #α_{q} & #β_{q} & #Γ & e & #(q − 1) & #α_{q} & #β_{q} & #Γ \\ 1 & 1 & 1 & 1 & 29 & 2 & 1 & 1 & 1 & 177 \\ 3 & 1 & 2 & 1 & 2411 & 4 & 2 & 2 & 2 & 5 \\ 5 & 2 & 1 & 2 & 375 & 6 & 1 & 2 & 2 & ≥ 6.10^{6} \\ 7 & 3 & 2 & 3 & 14 & 8 & 1 & 2 & 2 & ≥ 1.10^{9} \\ 9 & 1 & 1 & 3 & ≥ 1.10^{10} & 10 & 3 & 3 & 3 & 15 \\ 11 & 2 & 3 & 2 & 1 & 12 & 3 & 2 & 3 & 8 \\ 13 & 3 & 3 & 3 & 5 & 14 & 3 & 1 & 2 & ≥ 3.10^{7} \\ 15 & 1 & 3 & 2 & ≥ 2.10^{17} & 16 & 4 & 3 & 3 & 15 \\ 17 & 4 & 3 & 3 & 17 & 18 & 2 & 2 & 3 & 1 \\ 19 & 4 & 4 & 5 & 22 & 20 & 2 & 3 & 2 & 1 \\ 21 & 3 & 3 & 2 & 1 & 22 & 6 & 5 & 5 & 23 \\ 23 & 3 & 3 & 1 & ≥ 7.10^{12} & 24 & 2 & 4 & 2 & 6 \\ 25 & 5 & 4 & 5 & 23 & 26 & 3 & 2 & 2 & 1 \\ 27 & 6 & 3 & 3 & 20 & 28 & 4 & 4 & 4 & 19 \\ 29 & 2 & 2 & 5 & 1 & 30 & 1 & 2 & 4 & ≥ 1.10^{35} \\ 31 & 6 & 5 & 5 & 22 & 32 & 3 & 5 & 4 & 20 \\ 33 & 2 & 2 & 4 & 1 & 34 & 4 & 4 & 4 & 17 \\ 35 & 3 & 4 & 2 & 1 & 36 & 3 & 4 & 1 & ≥ 3.10^{20} \\ 37 & 7 & 7 & 5 & 27 & 38 & 4 & 4 & 5 & 20 \\ 39 & 3 & 3 & 1 & ≥ 1.10^{23} & 40 & 6 & 5 & 5 & 9 \\ 41 & 2 & 3 & 3 & 1 & 42 & 3 & 4 & 6 & 22 \\ 43 & 6 & 5 & 2 & 22 & 44 & 1 & 2 & 2 & ≥ 5.10^{51} \\ 45 & 5 & 5 & 5 & 24 & 46 & 3 & 5 & 4 & 18 \\ 47 & 5 & 5 & 4 & 24 & 48 & 2 & 5 & 4 & 1 \\ 49 & 8 & 6 & 6 & 28 & 50 & 2 & 3 & 3 & 1 \\ 51 & 2 & 3 & 3 & 1 & 52 & 10 & 7 & 7 & 54 \\ 53 & 1 & 3 & 3 & ≥ 4.10^{62} & 54 & 2 & 3 & 3 & 1 \\ 55 & 6 & 6 & 4 & 25 & 56 & 5 & 4 & 1 & ≥ 2.10^{32} \\ 57 & 6 & 6 & 6 & 30 & 58 & 9 & 9 & 7 & 36 \\ 59 & 6 & 5 & 7 & 30 & 30 & 4 & 3 & 3 & 8 \end{matrix}

Table 12: The number of prime divisors of q−1, α_{q} and β_{q}, and the number of non-isomorphic geometries of rank 2.

Corollary 7.1. If 2e + 1 is a prime number and q − 1, α_{q} and β_{q} are not prime, then the group Sz(q) has only one geometry of rank 2.

Proof. This is a direct consequence of theorem 7.1. □

Table 12 contains the number of prime divisors of q − 1 (#(q − 1)), α_{q} (#α_{q}), β_{q} (#β_{q}) and the number #Γ of non-isomorphic geometries obtained for every Suzuki group up to e = 60. Thanks to Corollary 7.1, we know that there is only one geometry for e = 11, 18, 20, 21, 26, 29, 33, 35, 41, 48, 50, 51 and 54.

8. A particular case: the Suzuki group Sz(8)

In this section we mention all the F, RC, FT, RWPRI geometries of rank 2 that can be constructed from Sz(8). These geometries have been determined using the theorem that has been demonstrated in this article. The diagrams have been studied further using MAGMA. Besides the numbering of each geometry, there is a number in parenthesis which indicates the number of non-conjugate geometries that are fused under the action of Aut(G). The classification of all the F, RC, FT, RWPRI geometries of Sz(8) (i.e. for all ranks) is available in [14].

Corollary 8.1. Up to isomorphism, there are 29 firm, residually connected geometries of rank 2 on which Sz(8) acts flag-transitively and residually weakly primitively. Their diagrams are given below.

Proof. Straightforward by theorem 7.1. □

[figure: 1. (3); 7; 2; 7; 6; 6; 2080; 2080; D_{14}; D_{14}; B = 2; RPRI]

[figure: 2. (3); 3. (3); 4. (3); 5. (3); 6. (1); 7; 3; 7; 6; 6; 2080; 2080; D_{14}; D_{14}; B = 2; RPRI]

[figure: 7. (3); 7; 4; 7; 6; 6; 2080; 2080; D_{14}; D_{14}; B = 2; RPRI]

[figure: 8. (3); 9. (3); 10. (3); 6; 2; 6; 12; 6; 2080; 1120; D_{14}; D_{26}; B = 2; QPRI]

[figure: 11. (3); 12. (3); 13. (1); 6; 3; 6; 12; 6; 2080; 1120; D_{14}; D_{26}; B = 2; QPRI]

[figure: 14. (3); 8; 2; 8; 4; 6; 2080; 2912; D_{14}; D_{10}; B = 2; QPRI]

[figure: 15. (3); 8; 3; 8; 4; 6; 2080; 2912; D_{14}; D_{10}; B = 2; QPRI]

[figure: 16. (3); 17. (3); 18. (3); 8; 4; 8; 4; 6; 2080; 2912; D_{14}; D_{10}; B = 2; QPRI]

[figure: 19. (1); 9; 3; 10; 4; 6; 2080; 2912; D_{14}; D_{10}; B = 2; QPRI]

[figure: 20. (3); 13; 5; 14; 1; 6; 2080; 7280; D_{14}; 2^{2}; B = 2]

[figure: 21. (3); 13; 5; 14; 1; 6; 2080; 7280; D_{14}; 4; B = 2]

[figure: 22. (1); 12; 6; 12; 1; 6; 2080; 7280; D_{14}; 4; B = 2]

[figure: 23. (1); 10; 5; 10; 7; 1; 2080; 520; D_{14}; 2^{3}:7; B = 7; (2T)_{1}; QPRI]

[figure: 24. (3); 5; 3; 5; 12; 12; 560; 560; 13:4; 13:4; B = 4; RPRI]

[figure: 25. (3); 6; 4; 6; 12; 4; 560; 1456; 13:4; 5:4; B = 4; RPRI]

[figure: 26. (1); 5; 3; 6; 12; 4; 560; 1456; 13:4; 5:4; B = 4; RPRI]

[figure: 27. (3); 8; 5; 8; 1; 12; 560; 3640; 13:4; 4 × 2; B = 4; QPRI]

[figure: 28. (3); 8; 5; 8; 4; 4; 1456; 1456; 5:4; 5:4; B = 4; (2T)_{1}; RPRI]

[figure: 29. (3); 14; 7; 14; 1; 4; 1456; 3640; 5:4; 4 × 2; B = 4; (2T)_{1}; QPRI]

9. Observations and future work

While computing the exact number of non-isomorphic geometries for every Sz(2^{2e+1}) with e ≤ 60, we observed the following facts.

• If Sz(q) does not have proper subgroups isomorphic to a Suzuki group, then Table 12 shows that its number of non-isomorphic geometries is either 1 or a number that grows very fast like 2411 for e = 3, more than 6 millions for e = 6, more than 10 billions for e = 9, etc.

• If Sz(q) has proper subgroups isomorphic to some Suzuki simple group, then its number

of non-isomorphic geometries is more than reasonable. This seems a little bit intriguing. We could have thought that when a Suzuki ovoid has subovoids, its collineation group should have more geometries. This is not the case.

Maybe we should add some condition in order to restrict the number of geometries that appear. If we impose (2T)_{1}, we see that when e > 2, there is only one geometry that survives this test (the case e = 1 gives 3 such geometries). Thus the huge number of geometries we obtained in these cases disappear and the number of geometries obtained is quite small.

Anyway, a geometry which seems to play a central role is the one given by the RWPRI configuration (Sz(q), D_{2(q−1)}, q : (q − 1), q − 1). It is the geometry that appears in all cases. It satisfies (2T)_{1}. This geometry will be studied further in another paper. Also, we intend to classify the rank > 2 RWPRI geometries of Sz(q) and we want to consider the P SL(2, q) because of their similarity with the Suzuki.

References

[1] Buekenhout, F.: The geometry of the finite simple groups. In: Ed. Rosati L.A., Buildings and the geometry of diagrams, 1181 (1986), 1–78.

[2] Buekenhout, F. (ed.): Handbook of Incidence Geometry. Elsevier, Amsterdam, 1995.

[3] Buekenhout, F.: About the history of the Suzuki groups. Preprint.

[4] Buekenhout, F.; Cara, P.; Dehon, M.: Geometries of small almost simple groups based on maximal subgroups. Preprint.

[5] Buekenhout, F.; Dehon, M.; Leemans, D.: An Atlas of residually weakly primitive geometries for small groups. To appear in Mem. Acad. Royale Belg., Classe des Sciences.

[6] Buekenhout, F.; Dehon, M.; Leemans, D.: All geometries of the Mathieu group M_{11} based on maximal subgroups. Experimental Mathematics 5 (1996), 101–110.

[7] Cannon, J.: CAYLEY: A language for group theory. Department of Pure Mathematics, University of Sydney, December 1982.

[8] Cannon, J.; Bosma, W.: Handbook of MAGMA functions. Department of Pure Mathematics, University of Sydney, November 1994.

[9] Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A.: Atlas of Finite Groups. Oxford U.P., 1985.

[10] Dehon, M.: Classifying geometries with CAYLEY. Journal of Symbolic Computation, 17 (1994), 259–276.

[11] Dehon, M.; Miller, X.: The RWPRI and (IP)_{2} geometries of M_{11}. In preparation.

[12] Gottschalk, H.; Leemans, D.: The residually weakly primitive geometries of the Janko group J_{1}. In: Editor A. Pasini et al., Groups and Geometries. Birkhäuser. Basel.

[13] Gottschalk, H.: A classification of geometries associated with P SL(3, 4). Diplomarbeit, Giessen, 1995.

[14] Leemans, D.: The residually weakly primitive geometries of the simple Suzuki group Sz(8). Preprint, 1997.

[15] Lüneburg, H.: Die Suzukigruppen und ihre Geometrien. Lecture Notes in Math. 10 (1965), Springer, Berlin 1965.

[16] Lüneburg, H.: Translation Planes. Springer, New York 1980.

[17] Suzuki, M.: On a class of doubly transitive groups. Ann. of Math. 75 (1962), 105–145.

[18] Tits, J.: Géométries polyédriques et groupes simples. Atti 2a Riunione Groupem. Math. Express. Lat. Firenze (1962), 66–88.

[19] Tits, J.: Ovoïdes et groupes de Suzuki. Arch. Math., 13 (1962), 187–198.

Received November 6, 1996; revised version February 24, 1997