Embeddings of Ree unitals in a projective plane over a field

We show that the Ree unital $\mathcal{R}(q)$ has an embedding in a projective plane over a field $F$ if and only if $q=3$ and $\mathbb{F}_8$ is a subfield of $F$. In this case, the embedding is unique up to projective linear transformations. Beside elementary calculations, our proof uses the classification of the maximal subgroups of the simple Ree groups.


Introduction
A t-(n, k, λ) design, or equivalently a Steiner system S λ (t, k, n), is a finite simple incidence structure consisting of n points and a number of blocks, such that every block is incident with k points and every t-subset of points is incident with exactly λ blocks. Let D = (P, B, I) be a design and Π = (P ′ , B ′ , I ′ ) a projective plane. The map ̺ : P ∪ B → P ′ ∪ B ′ is an embedding of U, provided it is injective, ̺(P) ⊆ P ′ , ̺(B) ⊆ B ′ , and ∀P ∈ P, ∀B ∈ B : P IB ⇔ ̺(P )I ′ ̺(B). The embedding ̺ is admissible, if for any autormorphism α of U, there is a collineation β of Π such that ̺(P α ) = ̺(P ) β holds for all P ∈ P.
An abstract unital or a unital design of order n is a 2-(n 3 + 1, n + 1, 1) design. The problem of the embeddings of abstract unitals in projective planes is a classical one with many old unsolved questions, see [3,4,5,10,11,12,13,15]. The classical hermitian unital H(q) of order q is constructed from the set of absolute points and non-absolute lines of the desarguesian plane PG(2, q 2 ). The abstract hermitian unital H(q) has a natural embedding in PG(2, q 2 ), which is unique up to projective equivalence, see [11,14]. Moreover, this embedding is admissible.
Another class of abstract unitals of order q = 3 2n+1 was discovered by Lüneburg [13]. Let Ree(q) = 2 G 2 (q) be the Ree group of order (q 3 + 1)q 3 (q − 1), q = 3 2n+1 , see [1,9]. Then Ree(q) has a 2-transitive action on q 3 + 1 points, namely by conjugation on the set of all Sylow 3-subgroups. The pointwise stabilizer of two points P, Q is cyclic of order q − 1 and thus contains a unique involution t. It follows that Ree(q) has a unique conjugacy class of involutions, and any involution t fixes exactly q + 1 points. The blocks of the Ree unital R(q) are the sets of fixed points of the involutions of Ree(q). R(q) admits the Ree(q) as a 2-transitive automorphism group; the full automorphism group is larger, for n ≥ 1, admitting also the field automorphism, see [3]. The smallest Ree unital R(3) and the smallest Ree group Ree(3) ∼ = PΓL(2, 8) ∼ = PSL(2, 8) ⋊ C 3 are a little different from the general case, see [2,6]. For q > 3, Ree(q) is simple.
Lüneburg [13] showed that the Ree unital R(q) has no admissible embeddings in projective planes of order q 2 (desarguesian or not). For q = 3, Grüning [6] proved that the smallest a Ree unital R(3) has no admissible embedding in any projective plane of order 9. Montinaro [15] extended these results by showing that for q = 3 and n ≤ q 4 , R(q) has no admissible embedding in a projective plane of order n. Moreover, if R(3) is embedded in a projective plane Π of order n ≤ 3 4 in an admissible way, then either Π ∼ = PG (2,8), or n = 2 6 , see [15,Theorem 5].
In this paper, we completely characterize the embeddings of R(3) in a projective plane over a field, extending Montinaro's result.
Theorem 1. Let F be a field and ϕ : R(3) → PG(2, F ) an embedding. Then the following hold: (i) F 8 is a subfield of F , and the image of ϕ is contained in a subplane of order 8.

(iii) The embedding is admissible.
Our main result is the following: Theorem 2. Let n be a positive integer, and q = 3 2n+1 . Suppose that Π is a projective plane such that for any embedding ϕ : R(3) → Π, the image ϕ(R (3)) is contained in a pappian subplane. Then the Ree unital R(q) has no embedding in Π.
In particular, R(q) has no embedding in a projective plane over a field.
These results suggest that the problem of projective embeddings of the Ree unitals can be reduced to the question whether the smallest Ree unital has an embedding in a non-desarguesian projective plane. This question is surprisingly hard, even if we assume that the embedding is admissible.
The structure of the paper is as follows. In section 2, we present the embedding of R(3) in PG (2,8), and some technical lemmas to ease the calculations in PG (2,8). In section 3, we study sets of five points of PG (2,8), determining ten external lines of a conic. Such external pentagons correspond to super O'Nan configurations of R(3); their properties are listed in section 4. In sections 5 and 6, we prove Theorems 1 and 2.

Preliminaries
The embedding of R(3) in PG(2, 8) deserves special attention. The construction was first given by Grüning [6], who attributes the idea to F.C. Piper. The embedding is slightly simpler to present in the dual setting. Let K be a non-singular conic in PG (2,8). The group of collineations leaving K invariant is The tangents of K have a common point N, which is called the nucleus of K (see [8]). The set O = K ∪ {N} is a hyperoval, that is, a set of 10 points such that each line intersects it in 0 or 2 points. The point P is external, if P ∈ O. The line ℓ is external, if ℓ ∩ O = ∅. There are 63 external points, 28 external lines, and each external point is incident with 4 external lines. In other words, the external points and the external lines form a (dual) unital U of order 3. Let G = PΓO (3,8) be the group of projective semilinear transformations of PG (2,8), preserving O. G is isomorphic to PΓL (2,8) and acts 2-transitively on the set of external lines. Hence, U has a 2-transitive automorphism group and R(3) ∼ = U by [2]. We call the isomorphism ϕ : R(3) → U a dual embedding of R(3) into PG (2,8) with respect to the conic K.
To make the computation in F 8 more transparent, we fix a root γ ∈ F 8 of the polynomial X 3 + X + 1 = 0 in F 8 . Then The trace map of F 8 over F 2 is defined as We fix the coordinate frame (X, Y, Z) in PG (2,8) and extend the action of the Frobenius automorphism Φ : x → x 2 to the points and lines of PG (2,8). In this way, we obtain a projective semilinear transformation of order 3. For c ∈ F 8 , the map is an elation with axis Z = 0.
(ii) is trivial. (iii) follows from the fact that in a finite field F q of even order, the quadratic form X 2 + mX + b is reducible if and only if Tr(b/m 2 ) = 0. (iv) and (v) are immediate.
Finally, we present a useful elementary result on groups acting on graphs: Lemma 4. Let G be a group acting primitively on the set of vertices of the graph Γ. Then Γ is either empty or connected.
Proof. The connected components of Γ are blocks of imprimitivity of G.
Definition 5. Let K be a conic in PG(2, q). We say that the points P 0 , . . . , P 4 in general position form an external pentagon with respect to K, if P i P j are external lines of K, 0 ≤ i < j ≤ 4. The external pentagon is said to have type A 4 , if there is a group G 0 of collineations preserving K and {P 0 , . . . , P 4 }, such that G 0 ∼ = A 4 .
Notice that the points of an external pentagon are not in K, and if q is even, then they are also distinct from the nucleus of K. Lemma 6. Let P = {P 0 , . . . , P 4 } be a external pentagon of type A 4 in PG(2, q) with respect to the conic K. Then q is even and the following hold: (i) G 0 fixes one point of P and acts 2-transitively on the remaining four.
(ii) If q = 8, then G 0 ∼ = A 4 is the full stabilizer of the sets K and P in the collineation group of PG (2,8).
. Hence, T acts faithfully on P, with a unique fixed point, say, P 0 . As T is normal in G 0 , G 0 fixes P 0 . Moreover, T acts regularly and G 0 acts 2-transitively on {P 1 , . . . , P 4 }. Let H be the stabilizer of P 0 in PΓO(3, q). If q is odd then H ∼ = D 2(q±1) ⋊ C e . The dihedral subgroup D 2(q±1) contains a unique central involution α, which is contained in each elementary abelian subgroup of order 4. Hence, no subgroup of H of order 12 can be isomorphic to A 4 , a contradiction.
(ii) Assume q = 8. Then H ∼ = F + 8 ⋊ C 3 has order 24 and we have to show that H does not leave P invariant. Let t 0 be the tangent line through P 0 to K. The elementary abelian part A of H consists of elations with respect to t 0 , that is, for any 2-element α ∈ A, the set of fixed point of α is t 0 . As for i ≥ 1, P i ∈ t 0 , and the A-orbit of P i has length 8.
Lemma 8. The fundamental pentagon F is an external pentagon with respect to the conic K : X 2 + Y Z = 0. Moreover, F is of A 4 type with collineation group G 0 = Γ, Φ .
Proof. The following facts can be checked by calculations: A is fixed by G 0 . Φ fixes C 1 and permutes C 1 , C 2 , C 3 . The Γ-orbit of C 1 is {C 1 , . . . , C 4 }. The lines AC 1 : Y = X + Z and C 1 C 2 : Y = γX + Z are external. Lemma 9. Let K be a conic in PG (2,8) and P an external pentagon of type A 4 with respect to K. The projective coordinate frame can be chosen such that K has equation X 2 + Y Z = 0 and P is the fundamental pentagon.
Proof. We can assume the equation K : X 2 + Y Z = 0 and that A(1, 1, 0) is the point of P which is fixed by G 0 . Then G 0 is a subgroup of which is the stabilizer of K and A. The 2-subgroup {τ c | c ∈ F 8 } has two Φ-invariant, irreducible proper subgroups: Z(H) = τ 1 and Γ. Hence, Γ, Φ is the unique subgroup H which is isomorphic to A 4 . G 0 = Γ, Φ follows. Let C 1 denote the point of P \ {A} that is fixed by Φ. As AC 1 is an external line, C 1 must have coordinates (x, x + b, 1) with x, b ∈ F 2 and Tr(b) = 1. This means that either C 1 = (0, 1, 1) or C 1 = (1, 0, 1). Applying τ 1 to P, we can assume C 1 = (0, 1, 1). Straightforward computation shows that {C 1 , . . . , C 4 } is the Γ-orbit of C 1 , and {A, C 1 , . . . , C 4 } is indeed the funtamental pentagon. For the rest of this section, we use the notation of Lemma 3 for K, Γ, G 0 , and τ c . Proposition 11. Let F = {A, C 1 , . . . , C 4 } be the fundamental pentagon in PG (2,8). For any even permutation ijkℓ of {1, 2, 3, 4}, let d ijkℓ denote the line connecting the points AC i ∩ C j C ℓ and AC j ∩ C k C ℓ . The following hold: (i) G 0 permutes the lines d ijkℓ regularly. In particular, the lines d ijkℓ are distinct.
(ii) The lines d ijkℓ are external to K.
(iii) For any coset Γg of Γ, the four lines d π , π ∈ Γg, share a common point at infinity Z = 0. (iv) The lines AC 4 , d 1234 and d 3241 are concurrent.
Proof. Obviously, G 0 acts on the lines d ijkℓ . We have is not fixed by any element of order 3 of G. Thus, the stabilizer of d 1234 in G is contained in Γ. Since Tr(γ 6 ) = 1, the stabilizer of d 1234 in Γ is trivial by Lemma 3(i). This proves (i), and also (ii), since d 1234 is an external line. (iii) follows from the fact that Γ fixes the points at infinity. Computing the equations and the determinant det    we obtain (iv).

Super O'Nan configurations in R(3)
In a 2-design, an O'Nan (or Pasch) configuration consists of four pairwise intersection blocks, no three of which pass through the same point. Brouwer [2] observed that in R(3), each O'Nan configuration is contained in a super O'Nan configuration, that is, in a set of five pairwise intersecting blocks in general position. In this section, we collect some facts on super O'Nan configurations of R(3).
By Proposition 14(ii), these are collinear points, thus, The difference of these two determinants is which implies u = v ± 1. Substituting back, we obtain either 2v − 2 = 0 or 2 = 0, which are not possible unless char(F ) = 2. In this case, all equations so far reduce to u + v + 1 = 0. Computing ϕ(a ∩ c 4 ) = (v, v + 1, 0) and This shows that u, v ∈ F 8 , and for any even permutation ijkℓ, ϕ(D ijkℓ ) is contained in the subplane PG (2,8). Hence, at least 22 points of ϕ(R(3)) is contained in PG (2,8). If Q is one of the remaining 6 points, then there are at least two blocks through Q with equation over F 8 , and therefore Q is in PG (2, 8) as well. The computation shows that up to the action of the Frobenius map Φ, the embedding ϕ is uniquely determined by the images of the blocks c 1 , . . . , c 4 . In particular, ϕ must be an embedding with respect to a dual conic K * . As all dual conics of subplanes of order 8 are projectively equivalent, we obtain (ii). Montinaro's [15,Theorem 5] implies (iii). Proof. Consider the dual embedding ϕ * : R(3) → PG(2, F ). By Theorem 1, ϕ * (R(3)) is contained in a subplane PG (2,8). As before, we identify the blocks of R(3) and the involutions of Ree (3). For a block a, ϕ * (a) is an external point of the conic K. Moreover, a determines a unique collineationâ of order 2. The following are equivalent: (1) Two involutions a 1 , a 2 ∈ Ree(3) commute.
(4)â 1 ,â 2 fix the same point of K. This implies that ϕ * (a 1 ), . . . , ϕ * (a 7 ) are contained in a tangent of K, which is our claim in the dual setting.
In order to be self-contained, we resume Kleidman's classification [9, Theorem C] of the maximal subgroups of G, see also [7]. If q ≥ 27 and H is a maximal subgroup of G, then one of the following cases occur: (M1) H is a 1-point stabilizer, isomorphic to the semidirect product of a group of order q 3 with the cyclic group of order q − 1. In Ree(q), the stabilizer of two points is cyclic of order q−1. Hence, the intersection of two Sylow 3-subgroups is trivial. This implies that any Sylow 3-subgroup S 0 of G 0 is contained in a unique Sylow 3-subgroup S of G, and S is left invariant by ψ α . Conversely, let S be a ψ α -invariant Sylow 3-subgroup of G. The normalizer N G (S) is a parabolic subgroup of G, isomorphic to the semidirect product of a group of order q 3 with the cyclic group of order q − 1. The centralizer of the field automorphism in N G (S) has order q 3 0 (q 0 − 1). This shows that S 0 = S ∩ G 0 is a Sylow 3-subgroup in G 0 .
Proposition 16. Let q 0 = 3 2n 0 +1 such that F q is an extension of F q 0 . Then R(q) has a subdesign D ∼ = R(q 0 ). Moreover, the stabilizer of D in Ree(q) is isomorphic to Ree(q 0 ). In particular, R(3) is a subdesign of R(q) with stabilizer subgroup Ree(3).
Proof. Remember that the points and blocks of R(q) can be identified with the Sylow 3-subgroups, and the involutions of Ree(q), respectively. Hence, any automorphism of G = Ree(q) induces an automorphism of R(q). The involutions fixed by ψ α and the Sylow 3-subgroups left invariant by ψ α form a subdesign D of R(q). As explained above, ψ α -invariant involutions and Sylow 3-subgroups of G correspond to involutions and Sylow 3-subgroups of G 0 . Hence, D ∼ = R(q 0 ). Let T 0 be the stabilizer of D in G; clearly G 0 ≤ T 0 . Looking at the list of maximal subgroups of G in [9, Theorem C], we see that either T 0 = Ree(q), or T 0 is contained in a subgroup isomorphic to Ree(q 1 ) with q 1 = q 1/β , β prime. Repeating this argument, we conclude that T 0 itself is isomorphic to a Ree group Ree(q * ), where F q * is a subfield of F q . As T 0 preserves the set of involutions of G 0 , the only possibility is q 0 = q * .
We are now in the position to prove Theorem 2.
Proof of Theorem 2. Let us suppose that an embedding ϕ : R(q) → Π exists. Let I denote the set of involutions of Ree(q). In three steps, we show that all lines ϕ(a) (a ∈ I) are concurrent, a contradiction.
Claim 2: For any a ∈ I, there is a point P a ∈ ϕ(a) of PG(2, F ) such that P a ∈ ϕ(b) for all b ∈ I with ab = ba.
Fix a ∈ I. The centralizer C G (a) is a × T , with T ∼ = PSL(2, q). T has a unique class J of involutions. For arbitrary b, c ∈ J, a, b, c is contained in a Sylow 2-subgroup of G. By claim 1, the lines ϕ(b), ϕ(c) intersect on ϕ(a). Fix b and define P a = ϕ(a) ∩ ϕ(b). As T acts primitively on J, Lemma 4 implies that for any c ∈ J, there are elements b 0 = b, b 1 , . . . , b k = c ∈ J such that b i b i+1 = b i+1 b i for all i = 0, . . . , k − 1. For all indices i, ϕ(a) ∩ ϕ(b i ) = ϕ(a) ∩ ϕ(b i+1 ). Hence, ϕ(a), ϕ(b) and ϕ(c) are concurrent. If c is an involution of C G (a), not in J ∪ {a}, then ac ∈ J and ϕ(a), ϕ(b) and ϕ(ac) are concurrent. Also, the lines ϕ(a), ϕ(c) and ϕ(ac) are concurrent, that shows P a ∈ ϕ(c).
Claim 3: All lines ϕ(a) (a ∈ I) of the embedding are concurrent. If a, b ∈ I commute then P a = P b . Fix arbitrary elements a, b ∈ I. By [9, Theorem C], G acts primitively on I. Hence by Lemma 4, there are elements a 0 = a, a 1 , . . . , a k = b such that a i a i+1 = a i+1 a i for all i = 0, . . . , k − 1. Then P a = P a 0 = . . . = P a k = P b , that finishes the proof.