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

DEMONSTRATIO MATHEMATICA Vol. 49 No 2 2016 Sara Arias-de-Reyna, Luis Dieulefait, Gabor Wiese CLASSIFICATION OF SUBGROUPS OF SYMPLECTIC GROUPS OVER FINITE FIELDS CONTAINING A TRANSVECTION Communicated by J. Smith Abstract. In this note, we give a self-contained proof of the following classification (up to conjugation) of finite subgroups of GSpnpF`q containing a nontrivial transvection for `≥ 5, which can be derived from work of Kantor: G is either reducible, symplectically imprimitive or it contains SpnpF`q . This result is for instance useful for proving ‘big image’ results for symplectic Galois representations. 1. 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 `≥5that 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 , PSpnpF`dq or PGSpnpF`dq occurs as a Galois group over the rational numbers for a positive density set of primes ` (cf. [ 2 ], [ 3 ], [ 1 ]). 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 [ 6 ], 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 [ 9 ] 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 [7]. 2010 Mathematics Subject Classification: 20G14. Key words and phrases: sympletic group over a finite field, transvection, classification of subgroups of linear groups. DOI: 10.1515/dema-2016-0012 c Copyright by Faculty of Mathematics and Information Science, Warsaw University of Technology Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 10/13/16 7:54 AM 130 S. Arias-de-Reyna, L. Dieulefait, G. Wiese 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 n -dimensional K -vector space V equipped with a symplectic form (i.e. nonsingular and alternating), denoted by xv, wy “ v‚w for v, w PV , is called a symplectic K -space. A K -subspace WĎV is called a symplectic K -subspace if the restriction of xv, wy to WˆW is nonsingular (hence, symplectic). The general symplectic group GSppV, x¨,¨yq “ : GSppVq consists of those APGLpVq for which, there is αPKˆ , the multiplier (or similitude factor) of A , such that pAvq‚pAwq “ αpv‚wq for all v, w PV . The multiplier of A is denoted by mpAq . The symplectic group SppV, x¨,¨yq “ : SppVq is the subgroup of GSppVq of elements with multiplier 1. An element τPGLpVq is atransvection if τ´idV has rank 1, i.e. if τ fixes a hyperplane pointwisely, and there is a line U such that τpvq ´ vPU for all vPV . 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 SppVq . Any symplectic transvection has the form Tvrλs P SppVq:uÞÑ u`λxu, vyv with direction vector vPVand parameter λPK(see e.g. [4], pp. 137–138). The main classification result of this note follows. A short proof, deriving it from [6], is contained in [3]. Theorem 1.1. Let K be a finite field of characteristic at least 5and V a symplectic K -vector space of dimension n . Then any subgroup G of GSppVq which contains a nontrivial symplectic transvection satisfies one of the following assertions: 1. There is a proper K-subspace SĂVsuch that GpSq “ S. 2. There are nonsingular symplectic K -subspaces SiĂV with i“ 1 , . . . , h of dimension m for some măn such that V“Àh i“1Si and for all gPG there is a permutation σgPSymhp the symmetric group on t 1 , . . . , huq with gpSiq “ Sσgpiq . Moreover, the action of G on the set tS1, . . . , Shu thus defined is transitive. 3. There is a subfield L of K such that the subgroup generated by the symplectic transvections of Gis conjugated pin GSppVqq to SpnpLq. 2. 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 Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 10/13/16 7:54 AM Classification of subgroups of symplectic groups. . . 131 group containing a transvection by the centres of the transvections in the subgroup. Let K be a finite field of characteristic ` and let V be a symplectic K -vector space of dimension n . Let G be a subgroup of GSppVq . The 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“SpnpLq of SpnpKq with L a subfield of K . Then the directions of the transvections of G form the L -vector space Ln contained in Kn . This is what we have in mind when we introduce the term pL, Gq -rational subspace below. In order to do so, we set up some more notation. Write LpGq for the set of 0 ‰vPV such that Tvrλs P G for some λPK . More naturally, this set should be considered as a subset of PpVq , 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 vPV, define the parameter group of direction vin Gas PvpGq:“ tλPK|Tvrλs P Gu. The fact that Tvrµs ˝ Tvrλs “ Tvrµ`λs shows that PvpGq is a subgroup of the additive group of K . If K is a finite field of characteristic ` , then PvpGq is a finite direct product of copies of Z{`Z . Denote the number of factors by rkvpGq . Because PλvpGq “ 1 λ2PvpGq for λPKˆ , it only depends on the centre U : “ xvyKPLpGq Ď PpVq , 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 Φ : VˆKpv,λqÞÑTvrλs ÝÝÝÝÝÝÝÝÑ tsymplectic transvections in SppVqu. The multiplicative group Kˆ acts on VˆK via xpv, λq : “ pxv, x´2λq . Passing to the quotient modulo this action yields a bijection pVzt0uˆKq{Kˆpv,λqÞÑTvrλs ÝÝÝÝÝÝÝÝÑtnontrivial symplectic transvections in SppVqu. When we consider the first projection πV : VˆKV modulo the action of Kˆwe obtain πV:pVzt0u ˆ Kq{KˆPpVq, which corresponds to sending a nontrivial transvection to its centre. Let W be a K-subspace of V. Then Φgives a bijection pWzt0u ˆ Kq{Kˆpv,λqÞÑTvrλs ÝÝÝÝÝÝÝÝÑ tnontrivial symplectic transvections in SppVqwith centre in Wu. Let L be a subfield of K . For a subset SĎV , we denote by xSyL the L -span Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 10/13/16 7:54 AM 132 S. Arias-de-Reyna, L. Dieulefait, G. Wiese of the elements of S inside V . We call an L -vector space WLĎV L -rational if dimKWK“dimLWL with WK : “ xWLyK and x¨,¨y restricted to WLˆWL takes values in L . An L -vector space WLĎV is called pL, Gq -rational if WL is L-rational and Φinduces a bijection pWLzt0u ˆ Lq{Lˆpv,λqÞÑTvrλs ÝÝÝÝÝÝÝÝÑ GX tnontrivial sympl. transvections in SppVqwith centre in WKu. Note that pWLzt0u ˆ Lq{Lˆis naturally a subset of pWKzt0u ˆ Kq{Kˆ. A typical example of pL, SpnpLqq -rational subspace can be constructed as follows: let VL be a symplectic L -vector space of dimension n , and consider the symplectic K -vector space V“VLbLK (where the symplectic form on VL is extended K -linearly to V ). Then any L -subvector space WL of VL is pL, SppVLqq-rational. A K -subspace WĎV is called pL, Gq -rationalisable if there exists an pL, Gq -rational WL with WK“W . We speak of an pL, Gq -rational symplectic subspace WL if it is pL, Gq -rational and symplectic in the sense that the restricted pairing is non-degenerate on WL . Let HL and IL be two pL, Gq - rational symplectic subspaces of V . We say that HL and IL are pL, Gq -linked if there is 0‰hPHLand 0‰wPILsuch that h`wPLpGq. 3. 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 LpGq 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 LĎK and an pL, Gq -rational symplectic plane HLĎV . For this it is necessary to replace G by one of its conjugates inside GSppVq . The main ingredient for the existence of pL, Gq -rational symplectic planes, which is treated in Section 5, is Dickson’s classification of the finite subgroups of PGL2pF`q. The next main step is to show that two pL, Gq -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 pL, Gq -rational symplectic plane further, using again the existence of ‘many’ transvections. We obtain a maximal pL, Gq -rational symplectic space ILĎV in the sense that LpGq Ă IKYIK K , which is proved in Section 7. The proof of Theorem 1.1 can be deduced from this (see Section 8) because either IK equals V , that is the Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 10/13/16 7:54 AM Classification of subgroups of symplectic groups. . . 133 huge image case, or translating IK by elements of G gives the decomposition in case 2. 4. Simple properties We use the notation from the Introduction. In this subsection we list some simple lemmas illustrating and characterising the definitions made above. Lemma 4.1. Let vPLpGq . Then xvyL is an pL, Gq -rational line if and only if PvpGq “ L. Proof. This follows immediately from that fact that all transvections with centre xvyKcan be written uniquely as Tvrλsfor some λPK. Lemma 4.2. Let WLĎV be an pL, Gq -rational space and UL an L -vector subspace of WL. Then ULis also pL, Gq-rational. Proof. We first give two general statements about L -rational subspaces. Let u1, . . . , ud be an L -basis of UL and extend it by w1, . . . , we to an L -basis of WL . As WL is L -rational, the chosen vectors remain linearly independent over K , and, hence, UL is L -rational. Moreover, we see, e.g. by writing down elements in the chosen basis, that WLXUK“UL. It is clear that Φsends elements in pULˆLq{Lˆ to symplectic transvections in G with centres in UK . Conversely, let Tvrλs be such a transvection. As WL is pL, Gq -rational, Tvrλs “ Turµs with some uPWL and µPL . Due to WLXUK“UL, we have uPULand the tuple pu, µqlies in ULˆL. Lemma 4.3. Let WLĎV be an L -rational subspace of V . Then the following assertions are equivalent: (i) WLis pL, Gq-rational. (ii) (a) TWLrLs:“ tTvrλs | λPL, v PWLu Ď Gand (b) for each UPLpGq Ď PpVq with UĎWK there is a uPUXWL such that PupGq “ L(i.e. xuyLis an pL, Gq-rational line contained in Uby Lemma 4.1). Proof. (i) ñ (ii). Note that (iia) is clear. For (iib), let UPLpGq with UĎWK . Hence, there is uPU and λPKˆ with Turλs P G . As WL is pL, Gq -rational, we may assume that uPWL and λPL . Lemma 4.2 implies that xuyLis an pL, Gq-rational line. (ii) ñ (i). Denote by ι the injection pWLzt 0 u ˆ Lq{Lˆ,Ñ pWKzt 0 u ˆ Kq{Kˆ . By (iia), the image of Φ ˝ι lies in G . It remains to prove the surjectivity of this map onto the symplectic transvections of G with centres in WK . Let Tvrλs be one such. Take U“ xvyK . By (iib), there is v0PU such that UL“ xv0yLĎWL is an pL, Gq -rational line. In particular, Tvrλs “ Tv0rµswith some µPL, finishing the proof. Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 10/13/16 7:54 AM 134 S. Arias-de-Reyna, L. Dieulefait, G. Wiese Lemma 4.4. Let APGSppVq with multiplier αPKˆ . Then ATvrλsA´1“ TAvrλ αs . In particular, the notion of pL, Gq -rationality is not stable under conjugation. Proof. For all wPV , ATvrλsA´1pwq “ ApA´1w`λpA´1w‚vqvq “ w` λpA´1w‚vqAv . Since A has multiplier α , w‚Av “αpA´1w‚vq , hence ATvrλsA´1pwq “ w`λ αpw‚AvqAv “TAvrλ αspwq. Lemma 4.5.The group Gmaps LpGqinto itself. Proof. Let gPG and wPLpGq , say Twrλs P G . Then by Lemma 4.4 we have gTwrλsg´1“Tgwrλ αs, where αis the multiplier of g. Hence, gpwq P LpGq. 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. Lemma 4.6. Let V be a symplectic K -vector space, 0 ‰u1, u2PV . If Tu1rλ1s´1Tu2rλ2sPta¨Id : aPKˆu, then Tu1rλ1s “ Tu2rλ2s. Proof. Assume that Tu1rλ1s´1Tu2rλ2s “ aId. Then for all vPV, Tu2rλ2spvq ´ Tu1rλ1spavq “ 0. In particular, taking v“u1, Tu2rλ2spu1q ´ Tu1rλ1spau1q “ u1`λ2pu1‚u2qu2´au1“0, hence either u1 and u2 are linearly dependent or a“ 1(thus both transvections coincide). Assume then that u2“bu1 for some bPKˆ . Then for all vPVwe have Tbu1rλ2spvq ´ Tu1rλ1spavq “ v`λ2b2pv‚u1qu1´av ´λ1apv‚u1qu1 “ pa´1qv` pλ2b2´aλ1qpv‚u1qu1“0. Choosing v linearly independent from u1 , we obtain a“ 1, as we wished to prove. 5. Existence of pL, Gq-rational symplectic planes Let, as before, K be a finite field of characteristic ` , let V be a n - dimensional symplectic K -vector space and let GĎGSppVq be a subgroup. We will now prove the existence of pL, Gq -rational symplectic planes if there are two transvections in Gwith nonorthogonal directions. Note that any additive subgroup HĎK can appear as a parameter group of a direction. Just take G to be the subgroup of GSppVq 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, Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 10/13/16 7:54 AM Classification of subgroups of symplectic groups. . . 135 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. [ 8 ]). 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 PGL2pF`q . Recall that an elation is the image in PGLpVqof a transvection in GLpVq. Proposition 5.1. Let V be a 2-dimensional K -vector space with basis te1, e2u and Γ ĎPGLpVq a subgroup that contains two nontrivial elations whose centers U1 and U2 are different. Let `m be the order of an ` -Sylow subgroup of Γ. Then K contains a subfield L with `m elements. Moreover, there exists APPGL2pKq such that AU1“ xe1yK , AU2“ xe2yK , and A Γ A´1 is either PGLpVLqor PSLpVLq, where VL“ xe1, e2yL. Proof. Since there are two elations τ1 and τ2 with independent directions U1 and U2 , Dickson’s classification of subgroups of PGL2pF`q (Section 260 of [ 5 ]) implies that there is BPPGL2pKq such that B Γ B´1 is either PGLpVLq or PSLpVLq , where L is a subfield of K . Note that the order of an ` -Sylow of Γ »PGL2pVLq (resp. Γ »PSL2pVLq ) coincides with the cardinality of L ; this implies that L has exactly `m elements. By Lemma 4.4, the direction of BτiB´1 is BUi for i“ 1 , 2and the lines BUi are of the form xdiyK with diPVL for i“ 1 , 2. As PSLpVLq acts transitively on VL , there is CPPSLpVLq such that CU1“ xe1yK and CU2“ xe2yK . Setting A : “CB yields the proposition. Although the preceding proposition is quite simple, the very important consequence it has is that the conjugated elations AτiA´1 both have direction vectors that can be defined over the same L-rational plane. Lemma 5.2. Let V be a 2-dimensional K -vector space, GĎGLpVq containing two transvections with linearly independent directions U1 and U2 . Let `mbe the order of any `-Sylow subgroup of G. Then K contains a subfield L with `m elements and there are APGLpVq and an pL, AGA´1q -rational plane VLĎV . Moreover, A can be chosen such that AUi“Ui for i“ 1 , 2. Furthermore, if u1PU1 and u2PU2 are such that u1‚u2PLˆ, then VLcan be chosen to be xu1, u2yL. Proof. We apply Proposition 5.1 with e1“u1 , e2“u2 , and Γthe image of G in PGLpVq , and obtain APGLpVq (any lift of the matrix provided by the proposition) such that A Γ A´1 equals PSLpVLq or PGLpVLq for the L -rational plane VL“ xu1, u2yLĎV , and AUi“Ui for i“ 1 , 2. For PSLpVLq and PGLpVLq it is true that the elations contained in them are precisely the images of Tvrλsfor vPVLand λPL. Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 10/13/16 7:54 AM 136 S. Arias-de-Reyna, L. Dieulefait, G. Wiese First, we know that all such Tvrλs are contained in SLpVLq and, thus, in AGA´1 (since A Γ A´1 is PSLpVLq or PGLpVLq ). Second, by Lemma 4.6 the image of Tvrλs in A Γ A´1 has a unique lift to a transvection in SLpVLq Ď AGA´1 , namely Tvrλs . This proves that the transvections of AGA´1 are precisely the Tvrλs for vPVL and λPL . Hence, VL is an pL, AGA´1q -rational plane. Lemma 5.3. Let U1, U2PLpGq be such that H“U1‘U2 is a symplectic plane in V . By G0 we denote the subgroup tgPG|gpHq Ď Hu and by G|H the restrictions of the elements of G0 to H . Then LpG|Hq Ď LpGq (under the inclusion PpHq Ď PpVq). Proof. Let τiPG be transvections with directions Ui for i“ 1 , 2. Clearly, τ1, τ2PG0 and their restrictions to H are symplectic transvections with the same directions. Consequently, Lemma 5.2 provides us with APGLpHq and an pL, AGA´1q-rational plane HLĎH. Let UPLpG|Hq . This means that there is gPG0 such that g|H is a transvection with direction U , so that Ag|HA´1 is a transvection in AG|HA´1 with direction AU by Lemma 4.4. As HL is pL, AG|HA´1q -rational, all transvections Tvrλs for vPHL and λPL lie in AG|HA´1 , whence AG|HA´1 contains SLpHLq . Consequently, there is hPAG|HA´1 such that hAU “ AU1 . But A´1hA PG|H , whence there is γPG0 with restriction to H equal to A´1hA. As γH ĎH, it follows that γU “γ|HU“A´1hAU “U1. Now, γ´1τ1γis a transvection in Gwith centre γ´1U1“U, showing UPLpGq. Corollary 5.4. Let U1, U2PLpGq be such that H“U1‘U2 is a symplectic plane in V . By G0 we denote the subgroup tgPG|gpHq Ď Hu and by G|H the restrictions of the elements of G0 to H . Then the transvections of G|Hare the restrictions to Hof the transvections of Gwith centre in H. Proof. Let T be the subgroup of G generated by the transvections of G with centre in H . We can naturally identify T with T|H . Let U be the subgroup of G|Hgenerated by the transvections of G|H. We have that T|HĂU. Apply Lemma 5.2 to the K -vector space H and the subgroup UĂGLpHq . There exists a subfield LĂK , and an L -rational plane HL such that U is conjugate to SLpHLq , hence U»SL2pLq . Applying Lemma 5.2 to the K -vector space H and the subgroup T|H , we obtain a subfield L1ĂK , and an L1 -rational plane HL1 such that T|H is conjugate to SLpHL1q , hence H»SL2pL1q . But LpT|Hq “ LpGq X H“LpG|Hq “ LpUq by Lemma 5.3, whence L“L1 and the cardinalities of U and T|H coincide. Therefore they are equal. Proposition 5.5. Let U1, U2PLpGq Ď PpVq which are not orthogonal. Then there exist a subfield L≤K , APGSppVq , and an L -rational symplectic Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 10/13/16 7:54 AM Classification of subgroups of symplectic groups. . . 137 plane HL such that AU1ĎHK , AU2ĎHK and such that HL is pL, AGA´1q - rational. Moreover, if we fix u1PU1 , u2PU2 such that u1‚u2PLˆ , we can choose HL“ xu1, u2yLand Asatisfying AU1“U1,AU2“U2. Proof. Let H“U1‘U2 and note that this is a symplectic plane. Define G0 and G|H as in Lemma 5.3. Lemma 5.2 provides us with BPGLpHq such that BUi“Ui for i“ 1 , 2and such that HL“ xu1, u2yL is an pL, BG|HB´1q - rational plane. We choose APGSppVq such that AH ĎH 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 HL is an pL, AGA´1q -rational symplectic plane in V. And, indeed, by Corollary 5.4, the nontrivial transvections of AGA´1 with direction in H coincide with the nontrivial transvections of BG|HB´1 , which in turn correspond bijectively to pHLzt0u ˆ Lq{L. Note that Theorem 1.1 is independent of conjugating G inside SppVq . Hence, we will henceforth work with pL, Gq -rational symplectic spaces (instead of pL, AGA´1q-rational ones). Corollary 5.6. (a) Let HL be an L -rational plane which contains an pL, Gq -rational line U1,L as well as an L -rational line U2,L not orthogonal to U1,L with U2,K PLpGq . Then HLis an pL, Gq-rational symplectic plane. (b) Let U1,L “ xu1yL be an pL, Gq -rational line and U2“ xu2yKPLpGq such that u1‚u2PLˆ . Then xu1, u2yL is an pL, Gq -rational symplectic plane. Proof. (a) Fix u1PU1,L and u2PU2,L such that u1‚u2“ 1, and call WL“ xu1, u2yL . Apply Proposition 5.5: we get LĎK and APGSppVq such that xAU1,LyK“ xu1yK , AU2“ xu2yK and WL is pL, AGA´1q -rational. Let a1, a2PKˆ be such that Au1“a1u1 and Au2“a2u2 . The proof will follow three steps: we will first see that Pu2pGq “ L , then we will see that HL satisfies Lemma 4.3(iia) and finally we will see that HL satisfies Lemma 4.3(iib). Let α be the multiplier of A . First note the following equality between α , a1and a2: 1“u1‚u2“1 αpAu1‚Au2q “ 1 αpa1u1‚a2u2q “ a1a2 α. Recall that PavpGq “ 1 a2PvpGq . From Lemma 4.4 we have PAvpAGA´1q “ 1 αPvpGq. On the one hand, since U1,L is pL, Gq -rational and u1PU1,L , we know that Pu1pGq “ L by Lemma 4.1. On the other hand, since xu1yL is pL, AGA´1q - rational, Pu1pAGA´1q “ L , hence Pu1pGq “ α a2 1 L . We thus have α a2 1 PL . Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 10/13/16 7:54 AM 144 S. Arias-de-Reyna, L. Dieulefait, G. Wiese SppI1,Lq – SpnpLq and we are in Case 3of Theorem 1.1. Hence, suppose now that I1,K ‰V. Either every gPG stabilises I1,K , and we are in Case 1and done, or there is gPG and vPI1,L with gpvq R I1,K . Set I2,L : “gI1,L . Note that I2,L ĎLpGq because of Lemma 4.4. Now we apply Corollary 8.2 to the decomposition V“I1,K ‘IK 1,K and obtain that gpI1,Kq Ď IK 1,K. Moreover LpGq “ LpgGg´1q Ď gI1,K YgIK 1,K “I2,K YIK 2,K. We now have LpGq Ď I1,K YI2,K YpI1,K ‘I2,KqK . Either I1,K ‘I2,K “V and pI1,K ‘I2,KqK“0, or there are two possibilities: • For all gPG , gI1,L ĎI1,K YI2,K . If this is the case, then G fixes the space I1,K ‘I2,K, and we are in Case 1, and done. • There exists gPG , vPI1,L such that gpvq R I1,K YI2,K . Set I3,L “gI1,L . Due to LpGq Ď I3,K YIK 3,K, we then have LpGq Ď I1,K YI2,K YI3,K Y pI1,K ‘I2,K ‘I3,KqK. Hence, iterating this procedure, we see that either we are in Case 1, or we obtain a decomposition V“I1,K ‘ ¨ ¨ ¨ ‘ Ih,K with mutually orthogonal symplectic spaces such that LpGq Ď I1,K Y ¨ ¨ ¨ Y Ih,K . Note that Lemma 8.1 implies that G respects this decomposition in the sense that for all iP t 1 , . . . , hu there is jP t 1 , . . . , hu such that gpIi,Kq “ Ij,K . If the resulting action of G on the index set t 1 , . . . , hu is not transitive, then we are again in case 1, otherwise in case 2. A. A result on transvections in a 3-dimensional vector space In this appendix, we provide a proof of the following result concerning subgroups in a 3-dimensional vector space that was used in Section 6: Proposition A.1. Let V be a 3-dimensional vector space over a finite field Kof characteristic `≥5, and let GĎSLpVqbe a subgroup satisfying: 1. There exists a 1-dimensional K-vector space Usuch that G|U“ tidUu. 2. There exist U1, U2, U3 three distinct centres of transvections in G such that UĘU1‘U2and U“ U3. Then pU1‘U2qXpU‘U3qis the centre of a transvection of G. This result is Theorem 3.1(a) of [ 10 ]. Below, we have written the proof in detail. We will essentially follow the original proof of Wagner [ 10 ], reformulating it with the terminology developed in this paper. We follow [ 8 ] when Wagner refers to the results proven there. We also used [ 9 ] to ‘get a feeling’ of the ideas used in [10]. Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 10/13/16 7:54 AM Classification of subgroups of symplectic groups. . . 145 The setting differs from that of the rest of the paper, since there is no symplectic structure. One consequence of this is that the axis of a transvection τ in SLpVq is not determined by its centre. Given any plane WĂV and any line UĂV , there exist transvections with axis W and centre U ; namely, fixing an element ϕPHompV, Kq “ V˚ of the dual vector space of V such that W“kerpϕq , and fixing a nonzero vector uPU , then all transvections in SLpVq with axis W and centre U are given by τpvq : “v`λϕpvqu for some λPK(cf. [4], p. 160). A key input in the proof is Lemma 5.2. In order to apply it to a subplane WĂV , we need to endow it with some symplectic structure. We do so by choosing any two linearly independent vectors e1 , e2 and considering the symplectic structure defined by declaring te1, e2uto be a symplectic basis. Proof of Proposition A.1. Without loss of generality, we may assume that G is generated by transvections. In particular, we may assume that GĎSLpVq. The hypotheses imply that the inclusion U3Ď pU1‘UqXpU2‘Uq does not hold. Indeed, assume that U3Ď pU1‘Uq X pU2‘Uq . We know that V“U1‘U2‘U , hence U1‘U“ U2‘U , so that pU1‘UqXpU2‘Uq has dimension 1. Therefore U3“ pU1‘UqXpU2‘Uq “ U , but by hypothesis U3“ U . Interchanging U1 and U2 if necessary we can assume that U3Ę U1‘U. For i“ 2 , 3, let W1,i “U1‘Ui and G1,i be the subgroup of GLpW1,iq generated by the transvections in G that preserve the plane W1,i . We want to endow W1,i with a suitable pL, G1,iq -rational structure. In particular, we want that these structures be compatible. For each i“ 1 , 2 , 3, fix a transvection TiPG with centre Ui . Note that, since G|U is the identity and U“ Ui , the axis of Ti (that is, the plane pointwise fixed by it) must be Ui‘U. The transvections T1 and T2 preserve the plane U1‘U2 , and since this plane does not coincide with the axis of T1 or T2 , they both act as nontrivial transvections on U1‘U2 . We apply Lemma 5.2 to the 2-dimensional K -vector space W1,2 (which we endow with a symplectic structure with symplectic basis tu1, u2u such that u1PU1 and u2PU2 ) and the group G1,2 and obtain a matrix APGL2pKq such that AU1“U1 , AU2“U2 and a subfield L of K such that pW1,2qL is an pL, AG1,2A´1q -rational plane. Since U is linearly independent from U1‘U2 , we can extend A to an element of GLpVq such that AU “U . Without loss of generality, we can replace G by AGA´1 and U3by AU3. Thus pW1,2qL“ xu1, u2yLis an pL, G1,2q-rational plane. Since V“U1‘U2‘U , we find a1, a2PK such that 0 ‰u`a1u1`a2u2P U3 with some uPU . By hypothesis a2‰ 0. Hence by normalising, we can Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 10/13/16 7:54 AM 146 S. Arias-de-Reyna, L. Dieulefait, G. Wiese assume 0‰u3:“ ´u`a1u1`u2PU3, so that we have the relation (1.1) u“a1u1`u2`u3. The set B“ tu1, u2, uu is a K -basis of V . The proof will be finished if we show that G contains a transvection of direction u3´u“ ´a1u1´u2P pU‘U3qXpU1‘U2q. Now we consider the plane W1,3 , and endow it with a symplectic structure with symplectic basis tu1, u3u . We claim that xu1, u3yL is an pL, G1,3q - rational plane. Indeed, if we show that xu1yL is an pL, G1,3q -rational line, then Corollary 5.6(b) applied to U1,L “ xu1yL and U3 (which lies in LpG1,3q because by hypothesis G contains a transvection with centre U3 ) yields the result. Consider the set of transvections of G with centre U1 . As discussed above, their axis is U‘U1“ tvPV : p2pvq “ 0 u , where p2 denotes the projection in the second coordinate with respect to the basis B . Thus any transvection of G with direction U1 can be written as T1pvq “ v`λp2pvqu1 for some λPK . Restricting T1 to W1,2 , and taking into account that p2pvq “ ´v‚u1 with vPW1,2 for the symplectic structure on W1,2 with symplectic basis tu1, u2u , it follows from the pL, G1,2q -rationality of xu1, u2yL that λPL . Now we restrict to W1,3 . Note that p2pvq “ v‚u1 for vPW1,3 , where ‚ denotes the symplectic structure on W1,3 defined by the symplectic basis tu1, u3u . Thus the restriction of T1 to W1,3 is T1pvq “ v`λpv‚u1qu1 . This proves the pL, G1,3q-rationality of xu1yL. The discussion above shows that, if we fix the basis tu1, uiu of W1,i , then G1,i contains SL2pLq ; in particular it contains the reflection given by pu1ÞÑ ´u1, uiÞÑ ´uiq . Since G acts as the identity on U , we obtain that G contains the element δ1,i given by pu1ÞÑ ´u1, uiÞÑ ´ui, u ÞÑ uq . With respect to the basis B, these elements have the shape δ1,2“¨ ˚ ˝ ´100 0´1 0 0 0 1 ˛ ‹ ‚and δ1,3“¨ ˚ ˝ ´100 0´1 0 0 2 1 ˛ ‹ ‚. Thus T:“δ1,2δ1,3“¨ ˚ ˝ 1 0 0 0 1 0 0 2 1 ˛ ‹ ‚ is a transvection of centre U and axis U‘U1 . Since 2is invertible in F` , we Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 10/13/16 7:54 AM Classification of subgroups of symplectic groups. . . 147 can find kPZsuch that Tk“¨ ˚ ˝ 100 010 011 ˛ ‹ ‚. The transvection Tk˝T3˝T´kPG has direction Tkpu3q “ u3´u ; this is the transvection we were seeking. Acknowledgements. S. A.-d.-R. worked on this article as a fellow of the Alexander-von-Humboldt foundation. She thanks the Université du Luxembourg for its hospitality during a long term visit in 2011. She was also partially supported by the project MTM2012-33830 of the Ministerio de Economía y Competitividad of Spain. L. V. D. was supported by the project MTM2012-33830 of the Ministerio de Economía y Competitividad of Spain and by an ICREA Academia Research Prize. G. W. was partially supported by the DFG collaborative research centre TRR 45, the DFG priority program 1489 and the Fonds National de la Recherche Luxembourg INTER/DFG/12/10COMFGREP. S. A.-d.-R. and G. W. thank the Centre de Recerca Matemàtica for its support and hospitality during a long term visit in 2010. The authors thank the anonymous referee of [ 3 ] and Gunter Malle for suggesting the alternative proof of Theorem 1.1 based on Kantor’s paper [ 6 ], which is given in [3]. References [1] S. Arias-de-Reyna, L. Dieulefait, S. W. Shin, G. Wiese, Compatible systems of symplectic Galois representations and the inverse Galois problem III. Automorphic construction of compatible systems with suitable local properties, Math. Ann. 361(3) (2015), 909–925. [2] S. Arias-de-Reyna, L. Dieulefait, G. Wiese, Compatible systems of symplectic Galois representations and the inverse Galois problem I. Images of projective representations, Trans. Amer. Math. Soc., in press, 2016. [3] S. Arias-de-Reyna, L. Dieulefait, G. Wiese, Compatible systems of symplectic Galois representations and the inverse Galois problem II. Transvections and huge image, Pacific J. Math. 281(1) (2016), 1–16. [4] E. Artin, Geometric Algebra, Interscience Publishers, Inc., New York-London, 1957. [5] L. E. Dickson, Linear Groups: With an Exposition of the Galois Field Theory, with an introduction by W. Magnus, Dover Publications Inc., New York, 1958. [6] W. M. Kantor, Subgroups of classical groups generated by long root elements, Trans. Amer. Math. Soc. 248 (1979), 347–379. [7] S. Z. Li, J. G. Zha, On certain classes of maximal subgroups in PSpp 2 n, F q , Sci. Sinica Ser. A 25 (1982), 1250–1257. Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 10/13/16 7:54 AM 148 S. Arias-de-Reyna, L. Dieulefait, G. Wiese [8] H. H. Mitchell, Determination of the ordinary and modular ternary linear groups, Trans. Amer. Math. Soc. 12 (1911), 207–242. [9] H. H. Mitchell, The subgroups of the quaternary Abelian linear group, Trans. Amer. Math. Soc. 15 (1914), 379–396. [10] A. Wagner, Groups generated by elations, Abh. Math. Sem. Univ. Hamburg 41 (1974), 190–205. S. Arias-de-Reyna UNIVERSITÉ DU LUXEMBOURG FACULTÉ DES SCIENCES, DE LA TECHNOLOGIE ET DE LA COMMUNICATION 6, RUE RICHARD COUDENHOVE-KALERGI L-1359 LUXEMBOURG, LUXEMBOURG E-mail: [email protected] L. Dieulefait DEPARTAMENT D’ÀLGEBRA I GEOMETRIA FACULTAT DE MATEMÀTIQUES UNIVERSITAT DE BARCELONA GRAN VIA DE LES CORTS CATALANES, 585 08007 BARCELONA, SPAIN E-mail: [email protected] G. Wiese UNIVERSITÉ DU LUXEMBOURG FACULTÉ DES SCIENCES, DE LA TECHNOLOGIE ET DE LA COMMUNICATION 6, RUE RICHARD COUDENHOVE-KALERGI L-1359 LUXEMBOURG, LUXEMBOURG E-mail: gab[email protected] Received June 9, 2014; revised version December 12, 2014. Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 10/13/16 7:54 AM