Classification of subgroups of symplectic groups over finite fields containing a transvection

In this note we give a self-contained proof of the following classification (up to conjugation) of subgroups of the general symplectic group of dimension n over a finite field of characteristic l, for l at least 5, which can be derived from work of Kantor: G is either reducible, symplectically imprimitive or it contains Sp(n, l). This result is for instance useful for proving"big image"results for symplectic Galois representations.


Introduction
In this paper we provide a self-contained proof of a classification result of subgroups of the general symplectic group over a finite field of characteristic ℓ ≥ 5 that contain a nontrivial transvection (cf. Theorem 1.1 below).
The motivation for this work came originally from Galois representations attached to automorphic forms and the applications to the inverse Galois problem. In a series of papers, we prove that for any even positive integer n and any positive integer d, PSp n (F ℓ d ) or PGSp n (F ℓ d ) occurs as a Galois group over the rational numbers for a positive density set of primes ℓ (cf. [AdDW13a], [AdDW13b], [AdDSW13]). A key ingredient in our proof is Theorem 1.1. When we were working on this project, we were not aware that this result could be obtained as a particular case of some results of Kantor [Kan79], hence we worked out a complete proof, inspired by the work of Mitchell on the classification of subgroups of classical groups. More precisely, in an attempt to generalise Theorem 1 of [Mit14] to arbitrary dimension, one of us (S. A.-d.-R.) came up with a precise strategy for Theorem 1.1. Several ideas and some notation are borrowed from [LZ82].
We believe that our proof of Theorem 1.1 can be of independent interest, since it is self-contained and does not require any previous knowledge on linear algebraic groups beyond the basics.
In order to fix terminology, we recall some standard definitions. Let K be a field. An ndimensional K-vector space V equipped with a symplectic form (i.e. nonsingular and alternating), denoted by v, w = v•w for v, w ∈ V , is called a symplectic K-space. A K-subspace W ⊆ V is called a symplectic K-subspace if the restriction of v, w to W ×W is nonsingular (hence, symplectic). The general symplectic group GSp(V, ·, · ) =: GSp(V ) consists of those A ∈ GL(V ) such that there is α ∈ K × , the multiplier (or similitude factor) of A, such that we have (Av) • (Aw) = α(v • w) for all v, w ∈ V . The multiplier of A is denoted by m(A). The symplectic group Sp(V, ·, · ) =: Sp(V ) is the subgroup of GSp(V ) of elements with multiplier 1. An element τ ∈ GL(V ) is a transvection if τ −id V has rank 1, i.e. if τ fixes a hyperplane pointwisely, and there is a line U such that τ (v)−v ∈ U for all v ∈ V . The fixed hyperplane is called the axis of τ and the line U is the centre (or the direction). We will consider the identity as a "trivial transvection". Any transvection has determinant 1. A symplectic transvection is a transvection in Sp(V ). Any symplectic transvection has the form with direction vector v ∈ V and parameter λ ∈ K (see e.g. [Art57], pp. 137-138).
The main classification result of this note is the following. A short proof, deriving it from [Kan79], is contained in [AdDW13b].
Theorem 1.1. Let K be a finite field of characteristic at least 5 and V a symplectic K-vector space of dimension n. Then any subgroup G of GSp(V ) which contains a nontrivial symplectic transvection satisfies one of the following assertions: 1. There is a proper K-subspace S ⊂ V such that G(S) = S.
2. There are nonsingular symplectic K-subspaces S i ⊂ V with i = 1, . . . , h of dimension m for some m < n such that V = h i=1 S i and for all g ∈ G there is a permutation σ g ∈ Sym h (the symmetric group on {1, . . . , h}) with g(S i ) = S σg (i) . Moreover, the action of G on the set {S 1 , . . . , S h } thus defined is transitive.
3. There is a subfield L of K such that the subgroup generated by the symplectic transvections of G is conjugated (in GSp(V )) to Sp n (L). The authors thank the anonymous referee of [AdDW13b] and Gunter Malle for suggesting the alternative proof of Theorem 1.1 based Kantor's paper [Kan79], which is given in [AdDW13b].

Symplectic transvections in subgroups
Recall that the full symplectic group is generated by all its transvections. The main idea in this part is to identify the subgroups of the general symplectic group containing a transvection by the centres of the transvections in the subgroup.
Let K be a finite field of characteristic ℓ and V a symplectic K-vector space of dimension n. Let G be a subgroup of GSp(V ). A main difficulty in this part stems from the fact that K need not be a prime field, whence the set of direction vectors of the transvections contained in G need not be a K-vector space. Suppose, for example, that we want to deal with the subgroup G = Sp n (L) of Sp n (K) for L a subfield of K. Then the directions of the transvections of G form the L-vector space L n contained in K n . It is this what we have in mind when we introduce the term (L, G)-rational subspace below. In order to do so, we set up some more notation. Write More naturally, this set should be considered as a subset of P(V ), the projective space consisting of the lines in V . We call it the set of centres (or directions) of the symplectic transvections in G. For a given nonzero vector v ∈ V , define the parameter group of direction v in G as is a subgroup of the additive group of K. If K is a finite field of characteristic ℓ, then P v (G) is a finite direct product of copies of Z/ℓZ. Denote the number of factors by rk v (G). Because of P λv (G) = 1 λ 2 P v (G) for λ ∈ K × , it only depends on the centre U := v K ∈ L(G) ⊆ P(V ), and we call it the rank of U in G, although we will not make use of this in our argument.
We find it useful to consider the surjective map The multiplicative group K × acts on V × K via x(v, λ) := (xv, x −2 λ). Passing to the quotient modulo this action yields a bijection When we consider the first projection π V : V × K ։ V modulo the action of K × we obtain which corresponds to sending a nontrivial transvection to its centre. Let W be a K-subspace of V . Then Φ gives a bijection Let L be a subfield of K. We call an L-vector space We speak of an (L, G)-rational symplectic subspace W L if it is (L, G)-rational and symplectic in the sense that the restricted pairing is non-degenerate on W L . Let H L and I L be two (L, G)-rational symplectic subspaces of V . We say that H L and I L are (L, G)-linked if there is 0 = h ∈ H L and 0 = w ∈ I L such that h + w ∈ L(G).

Strategy
Now that we have set up all notation, we will describe the strategy behind the proof of Theorem 1.1, as a service for the reader. If one is not in case 1, then there are 'many' transvections in G, as otherwise the K-span of L(G) would be a proper subspace of V stabilised by G. The presence of 'many' transvection is used first in order to show the existence of a subfield L ⊆ K and an (L, G)-rational symplectic plane H L ⊆ V . For this it is necessary to replace G by one of its conjugates inside GSp(V ). The main ingredient for the existence of (L, G)-rational symplectic planes, which is treated in Section 5, is Dickson's classification of the finite subgroups of PGL 2 (F ℓ ).
The next main step is to show that two (L, G)-linked symplectic spaces in V can be merged into a single one. This is the main result of Section 6. The main input is a result of Wagner for transvections in three dimensional vector spaces, proved in Appendix A.
The merging results are applied to extend the (L, G)-rational symplectic plane further, using again the existence of 'many' transvections. We obtain a maximal (L, G)-rational symplectic space I L ⊆ V in the sense that L(G) ⊂ I K ∪ I ⊥ K , which is proved in Section 7. The proof of Theorem 1.1 can be deduced from this (see Section 8) because either I K equals V , that is the huge image case, or translating I K by elements of G gives the decomposition in case 2.

Simple properties
We use the notation from the Introduction. In this subsection we list some simple lemmas illustrating and characterising the definitions made above. Proof. This follows immediately from that fact that all transvections with centre v K can be written uniquely as T v [λ] for some λ ∈ K.
Lemma 4.2. Let W L ⊆ V be an (L, G)-rational space and U L an L-vector subspace of W L . Then U L is also (L, G)-rational.
Proof. We first give two general statements about L-rational subspaces. Let u 1 , . . . , u d be an L-basis of U L and extend it by w 1 , . . . , w e to an L-basis of W L . As W L is L-rational, the chosen vectors remain linearly independent over K, and, hence, U L is L-rational. Moreover, we see, e.g. by writing down elements in the chosen basis, that Then the following assertions are equivalent: As W L is (L, G)-rational, we may assume that u ∈ W L and λ ∈ L. Lemma 4.2 implies that u L is an (L, G)-rational line.
It remains to prove the surjectivity of this map onto the symplectic In particular, the notion of (L, G)-rationality is not stable under conjugation.
Lemma 4.5. The group G maps L(G) into itself.
, where α is the multiplier of g. Hence, g(w) ∈ L(G).
The following lemma shows that the natural projection yields a bijection between transvections in the symplectic group and their images in the projective symplectic group.
hence either u 1 and u 2 are linearly dependent or a = 1 (thus both transvections coincide). Assume then that Choosing v linearly independent from u 1 , we obtain a = 1, as we wished to prove.

Existence of (L, G)-rational symplectic planes
Let, as before, K be a finite field of characteristic ℓ, V a n-dimensional symplectic K-vector space and G ⊆ GSp(V ) a subgroup. We will now prove the existence of (L, G)-rational symplectic planes if there are two transvections in G with nonorthogonal directions.
Note that any additive subgroup H ⊆ K can appear as a parameter group of a direction. Just take G to be the subgroup of GSp(V ) generated by the transvections in one fixed direction with parameters in H. It might seem surprising that the existence of two nonorthogonal centres forces the parameter group to be the additive group of a subfield L of K (up to multiplication by a fixed scalar). This is the contents of Proposition 5.5, which is one of the main ingredients for this article. This proposition, in turn, is based on Proposition 5.1, going back to Mitchell (cf. [Mit11]). To make this exposition self-contained we also include a proof of it, which essentially relies on Dickson's classification of the finite subgroups of PGL 2 (F ℓ ). Recall that an elation is the image in PGL(V ) of a transvection in GL(V ).
Proposition 5.1. Let V be a 2-dimensional K-vector space with basis {e 1 , e 2 } and Γ ⊆ PGL(V ) a subgroup that contains two nontrivial elations whose centers U 1 and U 2 are different. Let ℓ m be the order of an ℓ-Sylow subgroup of Γ.
Then K contains a subfield L with ℓ m elements. Moreover, there exists A ∈ PGL 2 (K) such that Proof. Since there are two elations τ 1 and τ 2 with independent directions U 1 and U 2 , Dickson's classification of subgroups of PGL 2 (F ℓ ) (Section 260 of [Dic58]) implies that there is B ∈ PGL 2 (K) such that BΓB −1 is either PGL(V L ) or PSL(V L ), where L is a subfield of K with ℓ m elements. By Lemma 4.4, the direction of Bτ i B −1 is BU i for i = 1, 2 and the lines BU i are of the form d i K with d i ∈ V L for i = 1, 2. As PSL(V L ) acts transitively on V L , there is C ∈ PSL(V L ) such that CU 1 = e 1 K and CU 2 = e 2 K . Setting A := CB yields the proposition.
Although the preceding proposition is quite simple, the very important consequence it has is that the conjugated elations Aτ i A −1 both have direction vectors that can be defined over the same Lrational plane.
containing two transvections with linearly independent directions U 1 and U 2 . Let ℓ m be the order of any ℓ-Sylow subgroup of G.
Then K contains a subfield L with ℓ m elements and there are A ∈ GL(V ) and an (L, Proof. We apply Proposition 5.1 with e 1 = u 1 , e 2 = u 2 , and Γ the image of G in PGL(V ), and obtain A ∈ GL(V ) (any lift of the matrix provided by the proposition) such that AΓA −1 equals for v ∈ V L and λ ∈ L.
First, we know that all such . This proves that the transvections of AGA −1 are precisely the T v [λ] for v ∈ V L and λ ∈ L. Hence, V L is an (L, AGA −1 )-rational plane. Proof. Let τ i ∈ G be transvections with directions U i for i = 1, 2. Clearly, τ 1 , τ 2 ∈ G 0 and their restrictions to H are symplectic transvections with the same directions. Consequently, Lemma 5.2 provides us with A ∈ GL(H) and an (L, AGA −1 )-rational plane H L ⊆ H. Let U ∈ L(G| H ). This means that there is g ∈ G 0 such that g| H is a transvection with direction U , so that Proof. Let H = U 1 ⊕U 2 and note that this is a symplectic plane. Define G 0 and G| H as in Lemma 5.3. Lemma 5.2 provides us with B ∈ GL(H) such that BU i = U i for i = 1, 2 and such that H L = u 1 , u 2 L is an (L, BG| H B −1 )-rational plane. We choose A ∈ GSp(V ) such that AH ⊆ H and A| H = B (this is possible as any symplectic basis of H can be extended to a symplectic basis of V ).
We want to prove that H L is an (L, AGA −1 )-rational symplectic plane in V . Note that Theorem 1.1 is independent of conjugating G inside Sp(V ). Hence, we will henceforth work with (L, G)-rational symplectic spaces (instead of (L, AGA −1 )-rational ones).
Corollary 5.6. (a) Let H L be an L-rational plane which contains an (L, G)-rational line U 1,L as well as an L-rational line U 2,L not orthogonal to U 1,L with U 2,K ∈ L(G).
Then H L is an (L, G)-rational symplectic plane.
Then u 1 , u 2 L is an (L, G)-rational symplectic plane.
Proof. (a) Fix u 1 ∈ U 1,L and u 2 ∈ U 2,L such that u 1 • u 2 = 1, and call W L = u 1 , u 2 L . Apply Proposition 5.5: we get L ⊆ K and A ∈ GSp(V ) such that AU 1,L K = u 1 K , AU 2 = u 2 K and W L is (L, AGA −1 )-rational. Let a 1 , a 2 ∈ K × be such that Au 1 = a 1 u 1 and Au 2 = a 2 u 2 . The proof will follow three steps: we will first see that P u 2 (G) = L, then we will see that H L satisfies Lemma 4.3 (iia) and finally we will see that H L satisfies Lemma 4.3 (iib). Let α be the multiplier of A. First note the following equality between α, a 1 and a 2 : Recall that P av (G) = 1 a 2 P v (G), and, from Lemma 4.4 it follows that P Av (AGA −1 ) = 1 α P v (G). On the one hand, since U 1,L is (L, G)-rational and u 1 ∈ U 1,L , we know that P u 1 (G) = L by Lemma 4.1. On the other hand, since u 1 L is (L, AGA −1 )-rational, P u 1 (AGA −1 ) = L, hence P u 1 (G) = α a 2 1 L. We thus have α a 2 1 ∈ L. Moreover, since u 2 L is (L, AGA −1 )-rational (e.g. using Lemma 4.2), we have that P u 2 (AGA −1 ) = L, hence P u 2 (G) = α a 2 2 L = a 2 1 α α 2 L = a 2 1 α L = L. This proves that u 2 L is (L, G)-rational by Lemma 4.1.
Next we will see that T H L [L] ⊆ G. Let b 1 , b 2 ∈ L with b 1 = 0 and λ ∈ L × . Consider the transvection T b 1 u 1 +b 2 u 2 [λ]. We want to prove that it belongs to G. We compute Note that since a 1 a 2 = a 2 1 α ∈ L and since W L = u 1 , u 2 L is (L, AGA −1 )-rational, it follows that AT b 1 u 1 +b 2 u 2 [λ]A −1 ∈ AGA −1 , and therefore T b 1 u 1 +b 2 u 2 [λ] ∈ G. Note that the same conclusion is valid for b 1 = 0 as u 2 L is (L, G)-rational.
Finally it remains to see that if U ∈ L(G) ∩ H L K , then there is u ∈ U ∩ H L with P u (G) = L. Assume that U ∈ L(G) ∩ H L K . Since we have seen that u 2 L is (L, G)-rational, we can assume that U = u 2 K . Therefore we can choose an element v ∈ U with v = u 1 + bu 2 , for some b ∈ K.
is a transvection with direction in L(AGA −1 ) ∩ W L , hence the (L, AGA −1 )-rationality of W L implies that b ∈ L.
(b) follows from (a) by observing that the condition u 1 • u 2 ∈ L × ensures that u 1 , u 2 L is an L-rational symplectic plane.
The next corollary says that the translate of each vector in an (L, G)-rational symplectic space by some orthogonal vector w is the centre of a transvection if this is the case for one of them.
Corollary 5.7. Let H L ⊆ V be an (L, G)-rational symplectic space. Let w ∈ H ⊥ K and 0 = h ∈ H L such that h + w K ∈ L(G). Then h 1 + w L is an (L, G)-rational line for all 0 = h 1 ∈ H L .
Proof. Assume first that H L is a plane. Letĥ ∈ H L withĥ • h = 1 (hence H L = h,ĥ L ). As ĥ L is an (L, G)-rational line andĥ • (h + w) = 1, it follows that ĥ , h + w L is an (L, G)-rational plane by Corollary 5.6. Consequently, for all µ ∈ L we have that µĥ + h + w L is an (L, G)-rational line. Let now µ ∈ L × . Then (µĥ + h + w) • h = µ = 0, whence again by Corollary 5.6 µĥ + h + w, h L is an (L, G)-rational plane. Thus, for all ν ∈ L it follows that µĥ + (ν + 1)h + w L is an (L, G)rational line. In order to get rid of the condition µ = 0, we exchange the roles of h andĥ, yielding the statement for planes.
Returning to the proof, if h 1 ∈ H L is nonzero, takeĥ ∈ H L such that h •ĥ = 0 and h 1 •ĥ = 0. First apply the Corollary to the plane h,ĥ L , yielding thatĥ + w is an (L, G)-rational line, and then apply it to the plane ĥ , h 1 L , showing that h 1 + w is an (L, G)-rational line, as required.
In the next lemma it is important that the characteristic of K is greater than 2.
Lemma 5.8. Let H L be an (L, G)-rational symplectic space. Let h,h ∈ H L different from zero and let w,w ∈ H ⊥ K such that w •w ∈ L × and h + w,h +w ∈ L(G). Then w,w L is an (L, G)-rational symplectic plane.
Proof. By Corollary 5.7 we have that h +w L is an (L, G)-rational line. As (h + w) • (h +w) = w •w ∈ L × , by Corollary 5.6 it follows that w −w L is an (L, G)-rational line. Since −h − w K ∈ L(G), by Corollary 5.7 we have that −h + w L is (L, G)-rational, and from (−h + w) • (h +w) = w•w ∈ L × we conclude that w+w L is an (L, G)-rational line. As (w−w)•(w+w) = 2w•w ∈ L × , we obtain that w +w, w −w L = w,w L is an (L, G)-rational symplectic plane, as claimed.

Merging linked orthogonal (L, G)-rational symplectic subspaces
We continue using our assumptions: K is a finite field of characteristic at least 5, L ⊆ K a subfield, V a n-dimensional symplectic K-vector space, G ⊆ GSp(V ) a subgroup. In the previous section we established the existence of (L, G)-rational symplectic planes in many cases (after allowing a conjugation of G inside GSp(V )). In this section we aim at merging (L, G)-linked (L, G)-rational symplectic planes into (L, G)-rational symplectic subspaces.
It is important to remark that no new conjugation of G is required. The only conjugation that is needed is the one from the previous section in order to have an (L, G)-rational plane to start from. Proof. The (L, G)-linkage implies the existence of h 1 ∈ H L and w 1 ∈ I L such that h 1 + w 1 K ∈ L(G). By Corollary 5.7 h + w 1 L is an (L, G)-rational line for all h ∈ H L . The same reasoning now gives that h + w L is an (L, G)-rational line for all h ∈ H L and all w ∈ I L .
In view of Lemma 4.3 the above is (iia). In order to obtain (iib), we need to invoke a result of Wagner. To make the exposition self-contained, we provide a proof in Appendix A.
Proposition 6.2. Let V be a 3-dimensional vector space over a finite field K of characteristic ℓ ≥ 5, and let G ⊆ SL(V ) be a group of transformations fixing a 1-dimensional vector space U . Let U 1 , U 2 , U 3 be three distinct centres of transvections in G such that U ⊆ U 1 ⊕ U 2 and U = U 3 . Then is the centre of a transvection of G. Proposition 6.3. Let U 1 , U 2 , U 3 ∈ L(G) and W = U 1 + U 2 + U 3 . Assume dim W = 3, U 1 and U 2 not orthogonal and let U be a line in W ∩ W ⊥ which is linearly independent from U 3 and is not Proof. Fix transvections T i ∈ G with centre U i , i = 1, 2, 3. These transvections fix W ; let H ⊆ SL(W ) be the group generated by the restrictions of the T i to W . The condition U ⊆ W ⊥ guarantees that the T i fix U pointwise. Note that furthermore U = U 3 and U ⊆ U 1 ⊕ U 2 . We can apply Proposition 6.2, and conclude that (U 1 ⊕ U 2 ) ∩ (U ⊕ U 3 ) is the centre of a transvection T of H. This transvection fixes the symplectic plane U 1 ⊕ U 2 . Call T 0 the restriction of T to this plane. It is a nontrivial transvection (since no line of U 1 ⊕ U 2 can be orthogonal to all U 1 ⊕ U 2 ). Hence by Lemma 5.3 the line (U 1 ⊕ U 2 ) ∩ (U ⊕ U 3 ) belongs to L(G).
We now deduce rationality statements from it. Corollary 6.4. Let H L be an (L, G)-rational symplectic plane and U 3 and U 4 be linearly independent lines not contained in H K . Assume U 4 ⊆ H K ⊕ U 3 is orthogonal to H K and to U 3 and assume that U 3 ∈ L(G).

Then the intersection H
Proof. Choose two (L, G)-rational lines U 1,L and U 2,L such that H L = U 1,L ⊕ U 2,L . With U = U 4 we can apply Proposition 6.3 in order to obtain that I : Proof. If necessary replacing H L by any (L, G)-rational plane contained in H L , we may without loss of generality assume that H L is an (L, G)-rational plane. Let y := h + w. If w = 0, the claim follows from the (L, G)-rationality of H L . Hence, we suppose w = 0. Then U 3 := y K is not contained in H K . Note that w is perpendicular to U 3 and to H K , and w ∈ H k ⊕ y K . Hence, Corollary 6.4 gives that the intersection H K ∩ (U 3 ⊕ w K ) = h K is in L(G).
Corollary 6.5 gives the rationalisability of a line. In order to actually find a direction vector for a parameter in L, we need something extra to rigidify the situation. For this, we now take a second link which is sufficiently different from the first link.
Corollary 6.6. Let H L ⊆ V be an (L, G)-rational symplectic space. Let 0 =h ∈ H K andw ∈ H ⊥ K such thath+w ∈ L(G). Suppose that there are nonzero h ∈ H L and w ∈ H ⊥ K such that h+w ∈ L(G) and w •w ∈ L × .

Extending (L, G)-rational spaces
We continue using the same notation as in the previous sections. Here, we will use the merging results in order to extend (L, G)-rational symplectic spaces.
Proposition 7.1. Let H L be a nonzero (L, G)-rational symplectic subspace of V . Let nonzero h,h ∈ H K , w,w ∈ H ⊥ K be such that h + w,h +w ∈ L(G) and w •w = 0. Then there exist α, β ∈ K × such that αw, βw L is an (L, G)-rational symplectic plane which is (L, G)-linked with H L .
Proof. By Corollary 6.5 we may and do assume by scaling h + w that h ∈ H L . Furthermore, we assume by scalingh +w that w •w = 1. Then Corollary 6.6 yields thath ∈ H L . We may appeal to Lemma 5.8 yielding that w,w L is an (L, G)-rational plane. The (L, G)-link is just given by h + w.
Proof. (a) Assume that g(v 1 ) ∈ S i and g(v 2 ) ∈ S j with i = j. Then g(v 1 ) + g(v 2 ) = g(v 1 + v 2 ) ∈ L(G) satisfies g(v 1 + v 2 ) ∈ S i ⊕ S j , but it neither belongs to S i nor to S j . This contradicts the assumption that L(G) ⊆ S 1 ∪ · · · ∪ S h .
(b) If S 1 = S 1,L with S 1,L an (L, G)-rational space, we can apply (a) to an L-basis of S 1,L .
Corollary 8.2. Let I L ⊆ V be an (L, G)-rational symplectic subspace such that L(G) ⊆ I K ∪ I ⊥ K and let g ∈ G. Then either g(I K ) = I K or g(I K ) ⊆ I ⊥ K ; in the latter case I K ∩ g(I K ) = 0.
Proof. This follows from Lemma 8.1 with S 1 = I K and S 2 = I ⊥ K .
We are now ready to prove Theorem 1.1.
Proof of Theorem 1.1. As we assume that G contains some transvection, it follows that L(G) is nonempty and consequently L(G) K is a nonzero K-vector space stabilised by G due to Lemma 4.5.
Hence, either we are in case 1 of Theorem 1.1 or L(G) K = V , which we assume now. From Proposition 5.5 we obtain that there is some A ∈ GSp(V ), a subfield L ≤ K such that there is an (L, AGA −1 )-rational symplectic plane H L . Since the statements of Theorem 1.1 are not affected by this conjugation, we may now assume that H L is (L, G)-rational.
From Corollary 7.4 we obtain an (L, G)-rational symplectic space I 1,L such that L(G) ⊆ I 1,K ∪ I ⊥ 1,K . If I 1,K = V , then we know due to I 1,L ∼ = L n that G contains a transvection whose direction is any vector of I 1,L . As the transvections generate the symplectic group, it follows that G contains Sp(I 1,L ) ∼ = Sp n (L) and we are in case 3 of Theorem 1.1. Hence, suppose now that I 1,K = V .
Either every g ∈ G stabilises I 1,K , and we are in case 1 and done, or there is g ∈ G and v ∈ I 1,L with g(v) ∈ I 1,K . Set I 2,L := gI 1,L . Note that I 2,L ⊆ L(G) because of Lemma 4.4. Now we apply Corollary 8.2 to the decomposition V = I 1,K ⊕ I ⊥ 1,K and obtain that g(I 1,K ) ⊆ I ⊥ 1,K . Moreover L(G) = L(gGg −1 ) ⊆ gI 1,K ∪ gI ⊥ 1,K = I 2,K ∪ I ⊥ 2,K . We now have L(G) ⊆ I 1,K ∪I 2,K ∪(I 1,K ⊕I 2,K ) ⊥ . Either I 1,K ⊕I 2,K = V and (I 1,K ⊕I 2,K ) ⊥ = 0, or there are two possibilities: • For all g ∈ G, gI 1,L ⊆ I 1,K ∪ I 2,K . If this is the case, then G fixes the space I 1,K ⊕ I 2,K , and we are in case 1, and done.
Hence, iterating this procedure, we see that either we are in case 1, or we obtain a decomposition V = I 1,K ⊕· · ·⊕I h,K with mutually orthogonal symplectic spaces such that L(G) ⊆ I 1,K ∪· · ·∪I h,K .
Note that Lemma 8.1 implies that G respects this decomposition in the sense that for all i ∈ {1, . . . , h} there is j ∈ {1, . . . , h} such that g(I i,K ) = I j,K . If the resulting action of G on the index set {1, . . . , h} is not transitive, then we are again in case 1, otherwise in case 2.
The transvections T 1 and T 2 preserve the plane U 1 ⊕ U 2 , and since this plane does not coincide with the axis of T 1 or T 2 , they both act as nontrivial transvections on U 1 ⊕ U 2 . We apply Lemma 5.2 to the 2-dimensional K-vector space W 1,2 (which we endow with a symplectic structure with symplectic basis {u 1 , u 2 } such that u 1 ∈ U 1 and u 2 ∈ U 2 ) and the group G 1,2 and obtain a matrix A ∈ GL 2 (K) such that AU 1 = U 1 , AU 2 = U 2 and a subfield L of K such that (W 1,2 ) L is an (L, AG 1,2 A −1 )rational plane. Since U is linearly independent from U 1 ⊕ U 2 , we can extend A to an element of GL(V ) such that AU = U . Without loss of generality we can replace G by AGA −1 and U 3 by AU 3 .
Since V = U 1 ⊕U 2 ⊕U , we find a 1 , a 2 ∈ K such that 0 = u+a 1 u 1 +a 2 u 2 ∈ U 3 with some u ∈ U . By hypothesis a 2 = 0. Hence by normalising, we can assume 0 = u 3 := −u + a 1 u 1 + u 2 ∈ U 3 , so that we have the relation u = a 1 u 1 + u 2 + u 3 . (1.1) The set B = {u 1 , u 2 , u} is a K-basis of V . The proof will be finished if we show that G contains a transvection of direction u 3 − u = −a 1 u 1 − u 2 ∈ (U ⊕ U 3 ) ∩ (U 1 ⊕ U 2 ). Now we consider the plane W 1,3 , and endow it with a symplectic structure with symplectic basis {u 1 , u 3 }. We claim that u 1 , u 3 L is an (L, G 1,3 )-rational plane. Indeed, if we show that u 1 L is an (L, G 1,3 )-rational line, then Corollary 5.6(b) applied to U 1,L = u 1 L and U 3 (which lies in L (G 1,3 ) because by hypothesis G contains a transvection with centre U 3 ) yields the result. Consider the set of transvections of G with centre U 1 . As discussed above, their axis is U ⊕ U 1 = {v ∈ V : p 2 (v) = 0}, where p 2 denotes the projection in the second coordinate with respect to the basis B. Thus any transvection of G with direction U 1 can be written as T 1 (v) = v + λp 2 (v)u 1 for some λ ∈ K. Restricting T 1 to W 1,2 , and taking into account that p 2 (v) = −v • u 1 with v ∈ W 1,2 for the symplectic structure on W 1,2 with symplectic basis {u 1 , u 2 }, it follows from the (L, G 1,2 )-rationality of u 1 , u 2 L that λ ∈ L. Now we restrict to W 1,3 . Note that p 2 (v) = v • u 1 for v ∈ W 1,3 , where • denotes the symplectic structure on W 1,3 defined by the symplectic basis {u 1 , u 3 }. Thus the restriction of T 1 to W 1,3 is T 1 (v) = v + λ(v • u 1 )u 1 . This proves the (L, G 1,3 )-rationality of u 1 L .
The discussion above shows that, if we fix the basis {u 1 , u i } of W 1,i , then G 1,i contains SL 2 (L); in particular it contains the reflection given by (u 1 → −u 1 , u i → −u i ). Since G acts as the identity on U , we obtain that G contains the element δ 1,i given by (u 1 → −u 1 , u i → −u i , u → u). Thus T := δ 1,2 δ 1,3 =    1 0 0 0 1 0 0 2 1    is a transvection of centre U and axis U ⊕ U 1 . Since 2 is invertible in F ℓ , we can find k ∈ Z such that T k =    1 0 0 0 1 0 0 1 1    . The transvection T k • T 3 • T −k ∈ G has direction T k (u 3 ) = u 3 − u; this is the transvection we were seeking.