V 10“! \ H95" " 63 m LL; «A LA. LIBRARY Michigan State University kw c This is to certify that the dissertation entitled LOCALLY FINITE SIMPLE LINEAR GROUPS presented by GARY M. SHUTE has been accepted towards fulfillment of the requirements for Ph . D. degree in Mathematics Zia/M 59%? Major professor Date AW! {26; /7S/Ol U / MS U i: an Affirmative Action/Equal Opportunity Institution 0— 12771 MSU LIBRARIES m \— RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES wiii be charged if book is returned after the date stamped beiow. ©1983 GARY MILTON SHUTE All Rights Reserved LOCALLY FINITE SIMPLE LINEAR GROUPS BY Gary Milton Shute A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1982 ABSTRACT LOCALLY FINITE SIMPLE LINEAR GROUPS BY Gary Milton Shute Let G be an infinite locally finite simple linear group. ‘We show that G is a Chevalley group over some locally finite field. This is proved by showing first'that G is an'ascending union of groups Gi' where the Gi are finite Chevalley groups with fixed Chevalley type and fixed characteristic. It is then shown that any such union is a Chevalley group of the same type and characteristic, defined over a locally finite field. As a consequence of this classification of locally finite simple linear groups, it is shown that an infinite locally finite simple group in which every prOper subgroup is solvable-by-finite is either PSL2(K) or Sz(K) for some infinite locally finite field K 'which contains no infinite proper subfields. Further. the fields K are precisely described as ascending unions of certain finite fields. ACKNOWLEDGMENTS I am deeply grateful ro Professor Richard Phillips for his guidance, patience, and encouragement in my work on this thesis. I would like to thank Professors Jonathan Hall and Brian Hartley for the several conversations which helped to shape this thesis. I am greatly appreciative of my wife, Linda Deneen for her moral support and encouragement. Finally, many thanks are due to Cindy Smith for her fast but careful typing. ii TABLE OF CONTENTS CHAPTER I. INTRODUCTION 1 0 1 ObjeCtives O O O O O O O O O O O O O O 1.2 Notation . . . . . . . . . . . . . . . . 1.3 Finite simple groups . . . . . . . . . . II. REDUCTIONS 2.1 Reduction of Theorem A to Theorem B . . 2.2 Regularity of embeddings . . . . . . . . 2.3 Reduction to Theoremc . . . . . . . . . III. PRELIMINARY LEMMAS 0N CHEVALLEY GROUPS IV. 3.1 3.2 3.3 3.4 3.5 The subgroups MR . . . . . . . . . . . The root subgroups of 1MK) . . . . . . General properties of the root subgroups The Cartan subgroups and diagonal automorphisms of iMK). . . . . . . . Commutator relations . . . . . . . . . . PROOF OF THEOREM C 4.1 4.2 4.3 4.4 4.5 Inner automorphisms . . . . . . . . . . Diagonal automorphisms . . . . . . . . . Graph automorphisms . . . . . . . . . . Field automorphisms . . . . . . . . . . Groups of types 2A21_1. Dz. 2E6 and all untwisted groups . . . . Groups of types 2A2t' 2B2, 2F4,. and 262 . . . . . . . . . . . . . . . . . Completion of the proof of Theorem C . . iii PAGE H 11 13 13 15 18 20 20 26 36 38 44 51 51 62 62 65 67 7O 78 CHAPTER PAGE V. THEOREM.D 81 5.1 Proof of Theorem D . . . . . . . . . . . . 81 5.2 Locally finite fields . . . . . . . . . . . 84 VI. CONCLUSION 91 6.1 Previous results towards Theorem B . . . . . 91 6.2 A conjecture on regular embeddings . . . . . 91 BIBLIOGRAPHY 95 iv INTRODUCT ION .;;l. In group theory, as in many other areas of mathematics, little progress can be made without additional assumptions beyond the basic axioms. One such assumption is that of finiteness, which in group theory has been exploited to considerable advantage, culminating in the recent classification of finite simple groups. In this thesis a weaker assumption is made. It is assumed that the groups being studied are locally finite, where a group G is locally finite if every finite subset of G is contained in a finite subgroup of G. Many of the results in the theory of locally finite groups have been extensions of theorems about finite groups, making use of the fact that a locally finite group is a direct limit of finite groups. This thesis is directed towards an extension of this kind. Specifically, the objective of this thesis is to apply knowledge about the finite simple groups to obtain information about locally finite simple groups. Ultimately, one h0pes to obtain a complete classification of locally finite simple groups. However, this is an aunbitious project which will probably not be completed for many years. One reason this task is so difficult is the 2 fact that direct limits of finite groups are not unique. In fact, it is known that there are 280 isomorphism classes of simple groups which can be obtained as ascending unions of finite alternating groups [5, Thm. 6.12]. This suggests that locally finite simple groups which contain arbitrarily large alternating groups, or perhaps even sections which are isomorphic to alternating groups, will be quite difficult to classify. On the other hand, there is a result of Kegel and Wehrfritz [5, Thm. 4.8] which together with the classification of finite simple groups shows that every locally finite simple group is either a linear group or it contains arbitrarily large alternating sections. In light of these facts, we have tempered our initial ambitions, aiming only for a classification of locally finite simple linear groups. A large number of locally finite simple linear groups can be constructed as Chevelley groups over locally finite fields. In addition, it is easy to show that any finite group has a faithful linear representation. Thus the class of locally finite simple linear groups includes the finite alternating groups, the cyclic groups of prime order, and the 26 "sporadic" finite simple groups, in addition to the Chevalley groups. The main result of this thesis is 3 Theorem A: Every infinite locally finite simple linear group is a Chevalley group over some locally finite field. Thus the class of locally finite simple linear groups consists of precisely those groups described above. Most of this thesis will be devoted to the proof of Theorem.A. In Chapter 2, two reductions are made towards the proof of Theorem A. First, the proof of Theorem A is reduced to that of Theorem B: An ascending union of a sequence of Chevalley groups of fixed type over finite fields with common characteristic p, is a Chevalley group of the same type over a locally finite field of characteristic p. The second reduction consists of showing that in an ascending sequence of groups satisfying the hypotheses of Theorem B, the embedding of one group into the next is "regular" (in a sense defined in Section 1.2) for all but a finite number of groups in the sequence. In Chapter 3, certain subgroups and sections of the Chevalley groups are examined in detail. Chapter 3 establishes notation and some basic lemmas for Chapter 4. Chapter 4 is devoted to the proof of Theorem C, a certain reduction needed in the proof of Theorem B. The precise statement of Theorem C is given in Section 1.2. In Chapter 5 an application of Theorem A is given: namely 4 Theorem D: If G is an infinite locally finite simple group in which every prOper section is solvable-by- finite then either G a PSL2(K) or G a Sz(K) 'where K is an infinite locally finite field which contains no infinite proper subfields. Also in Chapter 5, the class of locally finite fields is examined and classified, and precise descriptions of the fields K in Theorem D are obtained. .;;g. Notation. Our notation for the Chevalley groups is essentially that of [ 2 ]. We let ‘Kfi be the algebraic closure of the Galois field GF(p), where p is a prime. Corresponding to each simple complex Lie algebra Q, denote by Q(E§) the untwisted Chevalley group defined over 'Kp as described in [ 13 ]. The symbol a by itself is also used to denote the system of roots obtained from the Lie algebra t. For each root r 6 6, there is a homomorphism xr from the additive group (E%)+ onto the root subgroup Xr g Qtifi). The unipotent subgroup U is given by U = (Xr |r 6 §+>, where §+ is a positive system of roots in Q. B denotes the Borel subgroup of Qth). which can be identified as the normalizer of U in éth). The monomial subgroup is N = (nr(x) |r 6 Q4 x G'R;>, where nr(1) = xr(x)x_r(-k-1)xr(l) and 'K; denotes the multi- plicative group of non-zero elements in 'Efi. The Cartan subgroup is H = B n N. The Weyl group of affié) is the 5 section W = N/H. Finally, V denotes the subgroup (Xr |r 6 Q“), where Q- = {-r ‘r 6 5+}. All of the finite Chevalley groups can be described as subgroups of some §(E§) of the form . io . lUq) = (x e L! x |x q = X), where 1o is an auto- res r q morphism of 9(Kp) described in Table l, which is adapted from [ 2 , Table l]. The automorphism o = ioq permutes the set of subgroups [Xr |r 611}. This permutation is also called 0. The action of c on §(E%) is determined by its action on the elements xr(t) for azr EII where we have q Xr(t)o= )(t r). for tréII x olr The integers qr depend on o as well as r. This dependence is given in Table 1. As usual, in ioq, i denotes the order of the corresponding permutation on II: and i is deleted from the symbols ioq and ith) when this permutation is trivial. o N Nquuv .mum mom u Ho H~.H. o mnw& o N N . . U ooHouU . Nun HqUNumHuNqun..HHv Am. N16. H v o WlmfiYIO om N N H N o H vs m em a H so m m m w m m m am pm 6 U 0 v H¢.~clm.Hc om mfiIwIWMIIMIIw mm o v I: Q s I. s s H..He me ammo..e v ale m Ho on H am a HH.H-HV so vawuw... m m «a N qu N o N u o H on an m N G U H U0 mmww . . . II 9.0 m H m H . ‘ o m oouo.mum on UN N H. o m m othI.... m . e H Hue m w H m N H o H o m 0 «fl m N.H v ...HHuH.~cHH.HV o AVIAYu... m o N H Hue m m H H N a o H o m mcoHuUHHumwu Huo .u no H mm. .m. m m. Md. EnumMHn 5.5.55 m Mme :oHuom H H mHnme Table l (cont'd.) Powers of o All types (oq = o r q 2 . ‘ 2 r oqr 1f Z‘fr AL'DL'E6 ( Oq) = . o r 1f 2 [r q 20 if B‘fr 3 r qr D4 (Oq) = . o r 1f 3 ‘r q 2 . 0 1f 2 r B F G (20 )r= qr * 2' 4' 2 q o r/2 if 2 |r For the subgroups iMq) 'we can define subgroups U,B,N,H, and V by intersecting i{)(q) 'with the corresponding subgroups of afié). For the untwisted Chevalley groups (i = 1) this procedure also yields the root subgroups Xr of §(q). For the twisted groups the root subgroups are 0 XR = (x e H Xr |x = X) where R is an equivalence reR class of roots as described in [ 13, Section 13.2]. These subgroups are described in greater detail in Chapter 3. Throughout this thesis the following convention is observed. For Chevalley groups 1MK), 6 denotes the set g£_roots for the untwisted group §(K) and 8 II denotes £h_e §_e_1_:_ o_f_ fundamental 35292 E :5. TE s_e£ of; equivalence classes 9_n_ Q i_s denoted by Z , and th_e_ s31; 93 equivalence classes jwhigh contain fundamental £9235 g; Q is denoted by_ A. The set Z) is a root system in its own right, and A is a set of fundamental roots in 'ZL For both twisted and untwisted groups, W = N/H is isomorphic to the Weyl group W(Z) associated with Z. If K is a locally finite field of characteristic p for which the Chevalley group lMK) is defined, then J'atK) can be located as a subgroup of flip), Specifically we let 1MK) =lJ J'MKj) ‘where Kj is an ascending sequence 3 of finite subfields of K, as determined below. For untwisted Chevalley types, §(K) is defined for all fields K. If K is a locally finite field then the elements of K are algebraic over the prime field GFfp) so K g 5575 = 'Kp. Thus K is an ascending union of finite fields Kj = GF(qj). (see Chapter 5). The automorphisms o satisfy (0 )d = o , ensuring that MK.) 3 MK.) q q qd 1 j for i ng. For the Chevalley types 2821K) and 2F4’K), K has characteristic 2 and K admits an automorphism 8 satisfying 282 = 1. Then 9, restricted to a finite subfield of K, also satisfies 292 = 1. By [3, Inamma 14.1.1]. The finite subfields of K are fields 9 GF(2n) with n odd. Thus if K=UGFIKj) with J Kj = Gthj) then Ki is an odd degree extension of Ki for i g_j. As the automorphisms 2oq satisfy (20 )d = 20 d for d odd we have 2Q'Ki)$2§(Kj) for q i g_j. Similarly, we show that 2G2(Ki)g_2G2’Kj) when the finite fields Ki are as before. For the remaining twisted types l61K), K has an extension L of degree i over K. Let L = LJLj where J Lj = Gthj) is a finite subfield of L not contained in K. Let Kj = Lj n K: Then the degree of the extension (K3 :Kt) for t g.j is not divisible by i for that would imply that Lj g K‘. The automorphisms ioq satisfy (ioq)d = ioqd when i does not divide d so inj) g iuxz) for j g 1.. An argument similar to the proof of Lemma 3.1.1 below shows that the ascending unions constructed above are in fact isomorphic to the desired Chevalley group. Now, in Chapter 2 and Chapter 4, the arguments involve working with two or more Chevalley groups (usually of the same type) simultaneously. To avoid certain ambiguities and simplify notation two conventions are observed. First, the symbols 1MK) are reserved for the concrete groups as described above. Thus, if two groups l§(Kl) .and J'MKZ) appear together with 1§(Kl)g 3'MKZ), then 10 we have determined precisely how i§(K1) is embedded into i§(K2). Secondly, the structural subgroups Xr,U,B,N,H and V for an abstract Chevalley group G a iMK) are presumed to be the images of the correSponding subgroups of iMK) under an isomorphism from iMK) onto G. In contexts where either G or K is subscripted, the structural subgroups are also identified by a subscript. This applies for either abstract Chevalley groups or the concrete groups 1am). We are now in a position to define the term "regular" and than state Theorem C precisely. . . . i 1 Definition. Let Gl _ §(K1) and G2 _ §(K2) where K1, K have the same characteristic. An embedding 2 m :61 4 G2 is regglar if there is an element 9 e G2 such that 13? g 13923 and N? 3 N3. Theorem C: Let K1, K2 be finite fields with the same characteristic. Let G :i§(Kl) 4 i§(K2) be a regular embedding. Then i§(Kl)‘g_i§(K2) and m extends to an automorphism of iMKZ), except when i§(K1) = 2A2(2), 2 2 2 32(2). F4(2) or s2(3). Note that the conclusion 16(Kl) g l§(K2) implies that K2 is an extension of K1 whose degree is not divisible by i. 11 1.3. Finite simple groups. The finite simple groups are classified as in Table 2 below. For the purposes of this thesis the simple group 2F4l2)’ is included with the sporadic simple groups. Table 2 . Type A‘(q) 2Auq) B‘(q) 2B2(q) C‘(q) D£(q) 2D£(q) 3D4(q) E6(q) 2E6(q) E7(q) E8(q) F4(q) 2F4(q) Gz(q) 262(q) II. III. 12 The finite simple groups Chevalley groups Weyl group Sym(t+-1) 2”*1/2’18ym 2‘ Sym(t) lwl = 2 2‘ Sym(£) 2""1 Sym(t) 2‘-1 Sym(L-l) 22 Sym(2) |w| - 27-34-5 |W| = 27.32 ‘W‘ = 26.34.5. lWi = 214,35 |w| = 27-32 |w| = 24 |w| = 22.3 |w| = 2 Alternating groups: Common designation PSL‘+l(q) Psuz+1(q) P02‘+1(q) Sz(q) PSp2‘(q) PO;‘(q) PQ;‘(q) Alt(n), n'z 5. 27 Sporadic simple groups Restrictions £22ort=1,q24 12:2. (z.q)=+(2.2) 1,22, (Lq) + (2.2) q=2mm1 , mzil £12 3 1 2_4 t 2,4 II. REDUCTIONS 2.1. In this chapter we give the reduction of the proofs of Theorems A and B to that of Theorem C. Suppose G is an infinite locally finite simple linear group. By’[ 5, Prop. 4.6] G is an ascending union of finite simple groups, say G =|J Gj' To show that 3' Theorem B = Theorem A it suffices to show that there is a subsequence of {G1} which consists of Chevalley groups, all Of the same type and characteristic. This is made possible by the classification of finite simple groups, as listed in Table 2. First, we may dispose of any sporadic simple groups among the Gj as their number is finite. Now G g_GL(n,K) for some field K. By a well known theorem of Schur, any periodic linear group over a field of characteristic 0 contains an Abelian normal subgroup with finite index. Thus the characteristic of K must be some prime p. Then the p-subgroups of G have finite exponent [ ‘7, 2.6] so any p-section of G has finite exponent. This bounds the size of 13 14 the alternating groups which appear as sections of G. In particular only finitely many of the Gj are alternating groups so we may discard them. After eliminating the sporadic simple groups and the alternating groups, the groups Gj that remain must all be Chevalley groups. Each Chevalley group contains a section N/H which is isomorphic to its associated weyl group. All but a finite number of the Chevalley types fall into one of the infinite families At' B‘, CL' D‘, 2A‘, or 2D‘. For each of these types we see from table 2 that the Weyl group contains a section isomorphic to A1t(k) for some k.2 t/Z. Thus the rank parameter L must be bounded for the groups Gj' Consequently, only finitely many Chevalley types appear among the G. so one type, say 1Q, appears an infinite 3' number of times. That is, passing to a subsequence, we may assume that Gj a 1§(Kj) 'where IQ is a fixed Chevalley functor and {Kj] is a sequence of finite fields. We next show that there is an infinite subcollection of [Kj} ‘with uniform characteristic. Each of the groups Gj has a linear representation over Kj’ obtained by regarding G. as a group of automorphisms of the Lie 3 algebra of type t over Kj' As these representations all have the same degree, Malcev's theorem [ £5, 1.L.9] can be applied, showing G has a faithful linear 15 representation over any non-trivial ultraproduct of the fields Kj. If for each prime p there are only finitely many Kj ‘with characteristic p then a non-trivial ultra- product of the Kj has characteristic 0 . Then Schur's theorem implies that G is abelian-by-finite. As G is simple, there must be some prime p such that char(Kj) = p for infinitely many of the fields Kj. Passing again to a subsequence we may assume that G. a i§(Kj) and char(Kj) = p for all j, completing the 3 reduction to Theorem B. 2.2. We now wish to show that (2.2.1) Almost all of the embeddings j j+1 are regular. To prove 2.2.1 we first prove (2.2.2) For almost all Bj there is an element x» 3+1 xj-t-l EGj-I-l such that ngBj+1 . It is easy enough to produce an element xj+1 such x. 3+1 . that Uj g Uj+l since U3. and Uj+l are Sylow p-subgroups respectively. if Ujiil is the unigue of Gj and Gj+1 Sylow p-subgroup of Gj+1 which contains Uj then we have x. B. B.3+1 as well. To see this, let B. = N U.. :IS 3+1 96 3 G-J 3 16 . X . X . 9 3+1 9 . 3+1 9 _ 3+1 so U g_(Uj+l ) . By uniqueness, (Uj+1 ) - Uj+l , x. x. 3+1 = 3+1 8° 9 e N(:‘.j+1Uj4-1 Bj+1 ‘ Now G is a periodic linear group so by a theorem of‘Wehrfritz [ 4., Thm. Al] the maximal p-subgroups of G form a conjugacy class. By [ 5,. L.D.7, l.D.6, l.D.3], x. for almost every j, 053:1 is the unique conjugate of Uj+1 containing Uj' Thus, passing to a subsequence we x3+ 1 may suppose that for all j, Bj‘gBj+1 . Then for all XcX- coax. x o coco j. BjJ 3‘1 2 g ngili x2. By relabeling the structural subgroups of the Gj (conjugating Gj by xjxj_1°°°x2), ‘we may suppose that for all j, Bj S-Bj+l' Now, Hj+1 is a Hall p'-subgroup of the solvable group Bj+l' By Hall's theorem every p'-subgroup of Bj+1 is contained in some conjugate of Hj+l in Bj+l' In particular, for each j there is an element b. 3+1 . . bj+lte Bj+1 such that Hj‘gHj+1 . Again relabeling the structural subgroups of Gj, we may suppose that for all 3, Hj S-Hj+l' With the assumption (2.2.3) for large 3, Hj+1 is the unique Hall I . . p -subgroup 9£_ Bj+l containing Hj. We now show that (2.2.4) Nj _<. Nj+1 This involves two lemmas from Chapter 3, but the results in Chapter 3 do not depend in any way on the present chapter. 17 BY Lemma 3.4.1.we have N3+1 = N¢j+lHj. Thus to obtain 2.2.4, it suffices to show that Nj normalizes H Let neNj. Then Hj =HngBj+ an 3+1“ j+1' 5° is contained in some Hall p ’-subgroup L of n Bn By Lemma 3.4.2, the Hall p'-subgroups of j+1 I n nBj+1 are also Hall p -subgroups of both Bj+1 But by 2.2.3, Hj+1 is the unique Hall p’-subgroup of B containing Hj and Hn is the j+l j+1n unique Hall p'-subgroup of B2+l containing Hj = Hj' n Thus H. = L = H. and 3+1 so nEN 3+1 Gj+1Hj+1 = Nj+1' NjSNj+l as required. We now turn to the proof of 2.2.3. As the group = ngj is a linear group, B satisfies the minimal condition on centralizers. Thus for some n ‘we have CBHk= C BjH for all k,j 2 n. Suppose that 3.2 n and that Hj 3 all?” with b e 133.“. Let h (- Hj. Then [h,b] = h'lhb 6 Hj+l' On the other hand, Bj+l/Uj+1 a Hj+l which is Abelian so Bi+l'S'Uj+l' Thus [h, b] e Uj+ n 1“];1‘ . That is, b e CBHj. But then b 6 CBHj+:+1 so Hg+1 = Hj+1' Since the Hall p'-subgroups of Bj+l are conjugate in Bj+l' Hj+l must be the unique Hall p’-subgroup of Bj+l which contains Hj° This completes the proof of 2.2.1. Thus, passing to a subsequence, we may suppose that (2.2.5) For all j, the embedding Gj . Note that G is the subgroup of G generated by the. O unipotent elements of G that are fixed by a certain O automorphism o of G0. The automorphism o is described in [3, Sect. 13.4]. In each case 0 is a product of a field automorphism of G ‘with a graph automorphism O which normalizes Mo and so 0 normalizes ”0' Then MR is the subgroup of Mo generated by the unipotent elements of MO which are fixed by olMo. For the untwisted groups in (i), MO/C(MO) a AltL) by [3, Thm. 6.3.1] and olMo is a field automorphism with fixed field K. Thus it is easily shown that MR/C(MR):=A1(K). For case (iiL by [3, Thm. 6.3.1] and the Chevalley commutator relations [3, Thm. 5.2.2] M.O is isomorphic to 23 either YxY or YxYxY where Y/Q(Y) :- A1(l). As OIMO permutes the summands cyclicly, MR a- Y so MR/C (MR) a- A1(L). For the remaining cases, if R has type A2, B2, or G then Steinberg's characterization of the Chevalley 2 groups [3 , Thm. 12.1.1] shows that Mo/UMO) is isomorphic to A2(L), B2(K) or G2(K) respectively. It is easily verified that o|Mo is precisely the automorphism which defines the twisted groups 2A2(K), 2B2(K), or 2G2(K) respectively. Thus Lemma 3.1.1 holds in cases (iii), (iv), and (v), completing the proof. For the remainder of this section we suppose that K i_s _a_ locally finite field pf characteristic p. Also we adOpt the convention that ER = MR/UMR). For a2 element y 6 MR and a subgroup Y g ”11' 37 and T!- denote the images of y and Y, respectively, i_n ER. Now, for R e A, the set of fundamental roots of Z , let UR nixsls arts-#1:). V R n[xslse§3 , Sal-R]. By [3, Corr. 8.4.2 and Prop. 13.6.1], we have 3.1.2. U=URXR and V=VX for RGA. R -R' By [3, Thm. 8.5.2(i)], we have 24 3.1.3. MR normalizes U and V As the center of MR normalizes both XR and X_R we have by 3.1.2 and 3.1.3 that for R 6 A. C(MR) normalizes both U and V, so C(MR) g_H. Further, by conjugating by an element of N ‘we obtain 3.1.4. gunk) SH for all R 623. Since XR and X_R are p-groups and H is a p’-group it follows that 3plpg, For all R E A, the homomorphism.from MR onto ER is faithful on xR and x_R. Now, if K is finite we have lifil = lle' so an order argument shows that ii is a Sylow p-subgroup of iii. By an ascending union argument ‘ik is a maximal p-subgroup of ‘fik for K locally finite. Similarly KR is a maximal p—subgroup of ‘fifi. As Eh n i; = 1, 'we R may take iii and. NLR’ as the unipotent subgroups of ‘Efi. Further, we have Lemma 3.1.6. For all R 6 2L H n Mk is the Cartan subgroup, and N n MR is the monomial subgroup of ‘Mfi. .gpppgz Conjugating by an element of N, we may suppose without loss of generality that R e A. To show that 'fi_fi_M; is the Cartan subgroup of 'Ei we must show that 'H—fi—E; is the intersection of the normalizers of XR and X_R in MR. It is clear that H n MR normalizes 25 both lifi and ii;R since H normalizes both XR and X_R. Conversely, let 9 6 MR be such that '3 normalizes XR and X_R. Then 9 normalizes XRQ(MR) and X_Rg(Mh). As g(MR)‘g,H, XR and X_R are normal p-sub- groups of XRC(MR) and X_RQ(MR), respectively so 9 normalizes XR and x_ Thus by 3.1.2 and 3.1.3, 9 e H R. so 9 e H 0 MR as was to be shown. To show that N n MR is the monomial subgroup of Mk, let n1 6 Mk n N\H and let n2 6 MR be such that n2 is in the monomial subgroup of ER but .52 2’ H. We show that -l nln2 6 MR n H. Since 111 6 N, conjugation by 111 permutes the root subgroups. Since n1 6 Mk, but n1 (’H, it follows from n n1 - 3.1.2 and 3.1.3 that le = x and x R - —R XR. A similar —"2 - H52 argument shows that XR = X_R and X_R = are normal p-subgroups of XRg(MR) and ifi, and since S and X_ R R n2 n2 X_R§(Mh), respectively, we have XR = X_R and X_R = KR. -1 . . Consequently, nln2 normalizes both XR and X_R. Since -1 -l nln2 e Mk we have by 3.1.2 and 3.1.3 that nln2 e H, so nln;1 6 MR 0 H, completing the proof of Lemma 3.1.6. Corollary 3.1.7: Let u 6 XR\l. Then there are unique elements v’, v E X_ R and u E N n MR such that u = v'nv. 26 Proof: Since u a! 1, '3 a! 1 so 3 £354! MR n H since ‘ifi n ELR‘E;_fi_HIg.XR n X—RH = 1. For a Chevalley group G of rank 1, G = -VH U -VN_V' and the elements of VNV have a unique expression as a product 3’3; with V’, V E V and E E N. Thus there are unique elements 3’ , V E i—R and n 6 N n MR such that u = v’nv. Let v’, v be the unique preimages of '3'.'6 'which are contained in X_ and let R n1 be a preimage of '3. Then v'nlv ='E' so for some element 2 E C(MR) ‘we have u = zv'n, v a v‘znlv = v'nv, where n = znl. By 3.1.4 C(MR)‘g,H n Mk g_N n MR so n e N n MR. Since v’, v are uniquely determined, so is n = (v')'luv-1. Finally, we consider commutators of the form [x,h] with xexR and hEMRnH. If [x,h]-=1 thenwe have [.33] = [E75] = 1. Conversely, suppose that [E33] = 1. Then [x,h] 6 g(Mh)lg,H. But H normalizes XR so [x,h] E XR n H = 1. Thus we have proved Lemma 3.1.8: Let R E Z}. x e XR and h 6 MR n H. Then [x,h] a 1 if and only if [32.3”] =- 1. 3.2. The root subgroups of J'§(K). In establishing a notation for the elements of the root subgroups XR of 1MK) 'we need first of all to adOpt a convention for designating the elements of R. The elements of R constitute either a set of positive insets or a set of negative roots for the root system which 27 is spanned by R. We denote by ro.r1,... those elements of R which are fundamental roots or negatives of fundamental roots relative to the span of R. Other roots in R 'will be expressed as sums of the roots rj. Now the roots rj form an orbit under the action of the automorphism o = ioq which defines 'i§(K). ‘We choose the J labeling of the rj so that rj = r8 , with r a long 0 root if there are roots of different length. Let us call such a root, r0, the principal root of R. As the sum of long roots in Q is a long root we may suppose that for R, S 6 A, the principle root of Ri-S (which denotes the root of"Z ‘which contains sums rji-sk) is vet-so. This convention is for convenience in expressing commutator relations in section 3.4. Now the elements of XR are described in [3, Prop. 13.6.1] and multiplication formulas for the elements of SR are given in [3, PrOp. 13.6.2]. In [3], the elements of XR are described in terms of a field auto- morphism e sometimes denoted by a bar). In our notation e is the Frobenius automorphism of the extension field L (where L is defined in Section 3.1) defined as follows: 3.1.1 t6 = t r0, where rO is a principle root in R and qr is defined as in Table l for o = loq. O . From Table 1 it can be seen that a does not depend on R. 28 With notation amended from [3] to suit ourtpurposes, we list the known prOperties of the root subgroups xR and refine some of these prOperties in 3.2.1 - 3.2.4 below. The cases are determined by the Chevalley type of uR/cmR). 3.2.l. MR/C(MR) of type A1. In this case R is an equivalence class of type Al, A1 xAl, or A1 xAl xAl. ‘We define fields LR by {K if R has type Al : L if R has type Alel or Alelel . ‘We define elements xR(t) for t 6 LR by 3.2.1.1 x (t) for R of type A : r0 1 9 (t)= x (t)x (t) for R of type A xA ; xR r0 r1 1 1 9 62 xro(t)xrl(t )xr2(t ) for R of type Al xAl xAl . The multiplication rules in XR are, for all three types of R, 3.2.1.2 xR(t1)xR(t2) = XR(t1+t2) for tl.t2 6 LR. 3.2.2. MR/Q(MR) of type 2A2. In this case R is an equivalence class of type A2, containing roots re, and r +r . We let ‘1' o 1 29 LR = L . We define elements yR(t,u) by 3.2.2.1. yR(t,u) = xr (t)xr (t6)x (u) for t,u 6 LR o 1 J‘1‘”“2 such that u-t-ue = -N t9+1. r ,r O 1 Here, Nr r = i1 is a structure constant of the under- 0' l lying Lie algebra. By altering the Chevalley basis (replac1ng ero+rl by -erO+rl) _we may suppose that Then we have 6 3+1 3.2.2.2 xR = {yR(t.u) [t,u 6 LR' u+u = t 1. The elements of XR multiply according to _ 6 from which we obtain the commutator relations. - 6 6 We define the subgroup X2R by XZR = {yR(O,u) [u 6 LR, u+ue = O}. 3O Zmn . Now let q = p With n odd, and let 5 be a solution of the following equation over K: 2m 3.2.2.5 xp = l-x. 2"“+1 2m 2m .. Then 5" = (1--5)p =1-6p =5. Thus 2m+1 6 6 GF(p ) g_LR . 2mn 2m 2‘“(n—1) 2’“ Now 59 = 6" = 6" 6" = 6" since n-l is even. Thus we have by 3.2.2.5 69+6=l. For p=2 it is clear that 695(5. For p712. we may choose 6 so that 69 51 6. To see this, let m - 2 p 7’ O satisfy up +u = 0. Then both 5 and 5+“ satisfy 3.2.2.5. Since us a! [1. either 66 '74 6 or (6 + (1)9 a! 6 + u. Thus if we define e = 6 _59 then we have 6: a! 0. Further e satisfies 6 It is easy to show that every solution of x +x = O in LR is of the form at where t e K. Thus if we define 3.2.2.6 x2R(t) = yR(O,et) for t E K, hoe then have 31 3.2.2.7 x2R = {x2R(t) [t e K] We may further define elements 6+1) 3.2.2.8 xR(t) = yR(t,6t for t 6 LR.‘ Then we have 3.2.2.9 yR(t,u) = xR(t)x2R(e-l(u-6t)) 'whenever u+ue = t9+1 . Finally, by 3.2.2.4 it is easy to show that 3.2.2.10 XZR = [XR'XR] = C(XR) . 3.2.3. MR/I(Mh) of type 2B2. In this case R is an equivalence class of type B2, containing roots ro,r1,ro+-rl, and roi-Zrl. We let LR = K. The field K has characteristic 2, and the automorphism 9 satisfies 292=1. We define elements xR(t) and x2R(t) by 3.2.3.1 fin) = x (tentr (t)xr +r O O 1 II N x2R(t) (t)xr +2r (tze), for t 6 LR . O 1 O 1 32 Every element of KR can be uniquely expressed in the form yR( t,u) where yR(t.u) = xR(t)x2R(u),t,u 6 LR . Multiplication of elements in XR is given by .. 9 3.2.3.2 YR(tloul)YR(t20u2) "' YR(t1+t2.u1+u2+tlt2), from which we can derive the commutator relations. . __ 6 6 3.2.3.3 [yR(t1.ul).yR(t2.u2)] - yR(0,t1t24-tlt2). We also,have the identity 9+1) . 3.2.3.4 yR(t,u)2 = x(O,t We define the subgroup XZR by 3.2.3.5 XZR = [x2R(t) |t 6 LR]. Then by 3.2.3.4, we have _ 2 _ 3.2.3.6 XZR- [XGXRIX — l} . Further, from 3.2.3.2 we obtain 3.2.3.7 XZR g_ng . 3.2.4. MR/QUVIR) of type 2G2. In this case R has type G containing roots ro,r1,ro+rl, r +2r 2' O 1' and Zr +3r . We let I? +3r1, O l O The field K has characteristic 3, and the automorphism 9 satisfies 2 36 =1. we define elements xR(t), x2R(t), x3R(t) for t 6 LR by 6 6+1 29+l 3.2.4.1 (t) = x (t )x (t)x (t )x (t ) , -—-—-—- xR r1 r0 rO+rl ro+2r1 _ 6 x2R(t) - "raid-r1}t )xrd+3r1(t)' x3R(t) = xr +2r (te)x2r +3r (t)‘ O l O 1 Every element of XR can be uniquely expressed in the form yR(t,u,v), ‘where yR(t.u.v) = xR(t)x2R(u)x3R(v) t,u.v 6 LR . Multiplication of elements in XR is given by 39 = yR(t1+t2,ul+u2--tlt2 ) . 39+1 2 36 v1 + v2 - t2ul + tlt2 — tlt2 ) . from‘which we can derive the commutator relations 34 _ 39 36 39+1 2 3e 36 2 39+1 We also have the identity 3.2.4.4 yR(t.u.v)3= yR(o.o,-t3°+2) . We define the subgroups XZR and x3R by 3.2.4.5 XZR - {yR(O,u,v) |u,v 6 LR) , and .XBR - {x3R(t) |t 6 LR} . If we set t2 = O in 3.2.4.3. we obtain [yR(t1.u1.vl).yR(0.u2.v2)] = yR(o,o,tlu2) . from which it is easily shown that 3.2.4.6 XBR = [xR,x2R] = ng. From 3.2.4.4, we see that 3247 x ={xex[x3=1} .0. 2R R O This completes our case by case descriptions of the XR's. For convenience, we present in Table 3 information about the fields L and LR’ and the automorphisms e. we now move on to some more general pr0perties of the root subgroups. 35 N Hoe om N Hwy mm HEHa :35 H ANUV N EH2 H lea. a EH2 H HNHUV d E as HovH< m" M" “III. as .mmum m. M" "M m" Mt mum Mil.- Mm“ Nu" Hmoch anmuHmu E «a «mu HmovHa HuumnHmum EH2 m HH¢ H H+EN H+EN mnN q H+EN «Mme mZOHBUHMBmmm .ri SfiBmMm Boom QmBmHzB AHA WM“ HHN M ii zmsmwm Boom QmBmHZBZD Hove Abomw m WHMdH 36 3.3. General prOperties of the root subgroups. The groups XjR were defined in Sections 3.2.2, 3.2.3, and 3.2.4 for certain root subgroups KR. ‘We extend this definition'with 3.3.1 ij = 1 when j > 1' and Mh/Z(M§) has type A1, or when 3 > 2 and MR/C(MR) has type 2A2 or 232, or when 3 > 3 2 and MR/C(MR) has type G2. Under the same conditions we let ij(t) a l for t 6 LR. Then in all cases 3.3.2 XjR is characteristic in KR for 3.2 l. From the multiplication relations 3.2.1.2, 3.2.2.3, 3.2.3.2, and 3.2.4.2 it follows easily that Now let K1' K2 be-finite fields of characteristic p. Then the symbols ij(t) can be defined in either l3(K1) or in 16(K2). To circumvent this ambiguity we will distinguish all symbols defined relative to l6(K1) from those defined relative to 16(K2) by using subscripts. i . Thus, for example, ij,l(t) 6 §(K1) for all t 6 LR,1 and j 2_1. The following lemma permits us to ignore this distinction under certain conditions 37 Lemma 3.3.4: As elements of 6(Kp), ij,1(t) and ij'2(t) are equal for t e LR,1' prOVided that K1.S.K2 and the degree of K2 over K1 is relatively prime to i. Proof: In the untwisted groups this is clear. In the twisted groups of types 2A 2D . 2E6. the automorphisms 1' L 91 and 62 are characterized by the fact that ej is the unique automorphism of Lj of order 2, where Lj is the extension field of degree 2 over Kj. The condition that K2 is an odd degree extension of K thus ensures that 6 = 91. Then checking the definitions as 3.2.1.1, 3.2.2.1, 3.2.2.6, 3.2.2.8 verifies Lemma 3.3.4 for these groups. A similar argument can be made in groups of type B2 and 2F4, where ej is the unique automorphism of Kj which satisfies 26% = l, and in groups of type 2G where ej is the unique automorphism of Kj ‘which 2 2! satisfies 39; = l. 3 Finally, for groups of type D R is either of 4' type Al or type Alelel. For R of type A1, i o. o. (t)xr (t) 3x (t) 3 where Lemma 3.2.1 clearly holds. For R of type A xA1 xA 1 we note that x j(t) = x R' to o 1"o oj = 3oq . Suppose that q2 = q? ‘with (m,3) = 1. Then 3 . there is an integer a and an integer b = l or 2 such that m = 3a+b. Noting that (o | )3 = 1 and 1 3D (K ) 4 l _ m _ 3 that 02 — 01 from table 1, we have for t e LR,l - GF(ql) 38 c1 012 xr (t)xr (t) xr (t) if b = l, O O O (t) = xR,2 012 01 x (t)xr (t) xr (t) if b = 2. r0 0 0 Cl 012 Since ro and r0 are not joined in the Dynkin - o a diagram of D4, x (t) 1 and x (t) 1 commute so ‘0 J"o in either case we have xR 1(t) = xR 2(t), which completes the proof of Lemma 3.3.4. 3.4. The Cartan subgroups and diagonal automorphisms of J‘6(K). we begin this section with two lemmas required in Chapter 2. Lemma 3.4.1: Let GalMK) with |K|24. Then Proof: Suppose we can show that CUH = 1. Let g e NGH. Then for any h1 e H there is some h e H 2 such that hlg = gh2 . Now g can be expressed uniquely as a product I 9=U'nu. u EU. nEN. UEUnVn. Thus we have (u’) h nu II D" \Q E 39 But H normalizes U and U n Vn, and bin, nh2 e N so by uniqueness we have Since h can be any element of H, u’ e CUH = 1. Also 1 b can be any element of H so u e CUH = 1. Thus 2 g a n e N. To show that CUH = 1 'when |K|‘2_4, let u e CuH' ~As u e U, u can be expressed uniquely as a product u‘QYi' i where yi e XRi and [XRi] is a listing of the positive root subgroups of G in some specified order. Then for any h e H, h _ Eyi-nyi. Since H normalizes each root subgroup, y? 6 XR , so i by uniqueness, y? = yi for each i and each h e H. That is, each of the constituents, Yi' of u centralizes H. Thus to show that CUH = 1, it suffices to show that C H = l for all root subgroups XR. XR From the proofs of [3, Thm. 11.1.2 and Thm 14.4.1] it follows that CUH = l in each of the groups A1(K), 2A2(K), 2B2(K), and 2G2(K) whenever |K|12 4. Thus by Lemma 3.1.8, CXR(MR n H) = 1 whenever [Kl 2.4. As 4O CxRHSCmeR n H), ‘we have CXRH = l, completing the proof of Lemma 3.4.1. Lemma 3.4.2: Let G be a Chevalley group and let 9 e G. Then B n B9 contains a conjugate of H. Proof: Let g = blnb2 with n e N and bl,b2 e B. Then nb b b snsg=BnB 2=(Ban)22H2. The remainder of this section is devoted to lemmas used in the proof of Theorem C. ‘We consider first the diagonal automorphisms of J‘6(K), and let p be the characteristic of the finite field K. The elements of the group D of diagonal automorphisms of 6(Efi) can be identified with the group of R?-- characters on the group P of integral linear combinations of roots in n [3, Sect. 7.1]. For x E D and g e 6(Rfi), gx denotes the action of Xv regarded as an automorphism of 3(Kp) on 9. For r e 6, x(r) denotes the value of x, _as a Rfi-character of P, at r. By [3, p.100] we have -1 X - — Xr(t) — xr(x(r)t) for r E Z), t e KP. Now the Chevalley group 1@(K) is generated by its root subgroups XR for iR E A. For x e D and R 6 Z} it is easy to see from Section 3.2 that X normalizes XR if and only if the following conditions are satisfied: 41 3.4.3 i) x(ro) 6 LR 3' ii) x(rj) = x(rj)e for each root rj 6 R. Here the roots r0, rj, the field LR' and the automorphism 9 of LR are as defined in Section 3.2. For r E t, x(-r) = x(r)'l so X restricts to an automorphism of J'§(K) if and only if 3.4.3 is satisfied for all R e A. Let '5' denote the set of elements of D that restrict to automorphisms of iMK). For x 6‘5} ‘we may give meaning to the symbol x(R) by defining x(R) = x(ro) for R e 2; For R e A. the root r0 and the roots.(if any) rj are all of the fundamental roots of t. Thus an automorphism X 6'5, is determined by its values x(R) for R 6 {L It is easily verified that -1 3.4.4 xR(t)x = XR(x(R)t) for R e Z). t.e LR , x.€ D . And from 3.4.3 it follows that 3.4.5 For any set of elements [1R |R e A} with 1R 6 LR, there is a diagonal automorphism x of 1@(K) such that xR(t)X = xR(iRt) for R e A. t 6 LR . The following is also easily verified. 42 -1 3.4.6 i) X2R(t)x = x2R(x(R)9+1t), for R e A of type A2 or B2 and t E K. . -1 ii) x2R(t)x = x2R(x(R)3e+1t) and ; -l X3R(t)x = X3R(X(R)39+2t)o for R E A of type G2 and t e K. Now the subgroup H of i6(K) is a subgroup of the Cartan subgroup of §(EP) which can be regarded as a subgroup of D. H normalizes i6(K) and so for an element h e H, identified with a character x e D, x must satisfy 3.4.3. 3.4.4 and 3.4.6. Rather than working with the group H, it simplifies the proof of Theorem C if we work instead with the quotients HRH/CASH. The identification of HR as a subgroup of L; is via the map h 4 x(R) where x is the character associated with h. The following lemma is readily apparent from 3.4.4 and 3.4.6. Lemma 3.4.7: CHXR = NH(XXZR) for any element x e XR\X2R; further, F} CHKR = l. REA We finally have Lemma 3.4.8: |L£ :Hfil g_3 and IL; :HRI = 3 only if 14(K) = 2142m. 43 'gpppg: ‘We first show that IL; :th g_3. To do this we pass to the group 'Mfi = MR/C(MR). By 3;};3 we have for t 5 LR and h e H n “R h‘1 . . ———h—:f —————- xR(t) = XR(At) if and only if xR(t) = xR(xt). In view of Lemma 3.1.6, it suffices to show that |L; :HRI g_3 in the groups A1(K), 2A2(K), 232(K) or 2GZ(K). For these groups we find in [3, Lemma 14.1.2, where 11;:3“ is identified as d] that |I.;:H.Rl 1 or 2 2 for A1(K), 1 or 3 for 211200, and 1 for 132m) 2 * and G2(K). Thus ILR :HRI 3.3. Moreover, if lLR.:HR| = 3 then R has type A2 which occurs only if i6(K) = 2A2‘(K) for some 1. Conjugating by an element of N *we may suppose that R E A so that R is the root labelled R in Table 3. The A proof of Lemma 3.4.8 will be complete if we show that = * for 2 HRL LRL ‘With 1 > 1 ‘we may let. S be the root labelled A2£(K) with L > 1. R‘_1 in Table 3. Then S has type A1)6A1. Let s0 be the root in S: which is adjacent to the principle root in R, and let 3 be the other root in S . 1 To see that HR = L; , let p e L; and consider A L L the character = -1 .9, defined as in X“ xsotu xsltu [3, p. 98]. It can be verified that XH(RL) = u, and that x“ satisfies the hypotheses of [3, Thm. 13.7.2] 44 * so that xp 6 H. Thus HR = LR , completing the t L proof of Lemma 3.4.8. gpé, In this section we derive certain commutator relations between elements of root subgroups. We first consider roots, R, S E A ‘with R and S adjacent in the Dynkin diagram of A. If iMK) is an untwisted group then R = [v0] and S = [so] and by the Chevalley commutator relations [3, Thm. 5.2.2] we have 3.5.1 [x (u),xr (t)] = xr +s (N tu)y for t E K, 8o o o o S"o'ro where y denotes terms involving roots of height greater than 2, and N = i1. sooro If l§(K) is a twisted group, let RJWH==[rO,...,rm_1] and let S n n = [30,...,sn_1} with the roots ri and si as in Section 3.2. ‘We may suppose without loss of generality the diagram of (R u S) FIE is one of the following . 2 2 r ro :::>o S . 2 2 2 II r1 0 in A2‘_1, D‘, E6 r ° 3 III r1 30 in D4 r r o———o 3 IV 0 o in 2A1, 2E6 r1 H 81 r0 80 2 V in A2; r1 31 r O——( 8 o o . 2 VI (3 in F4 r1 0—H 31 For the purposes of this section we will compute only those contributions to [xs(u),xR(t)] made by terms from the root subgroups Xr +3 . Note that for ' k . of the roots R we have xR(t) = n x (t) and . r 3=O O n-l Ok all of the roots 3 we have xs(u) = ( U xS (u) )y k=0 0 y = 1 except in Cases V and VI. In Cases V and VI, y = x (*) so that y will have no effect on the so+sl terms from X Also note that in the formula rj+sk all for where 46 ab = a[a,b], if a is in one of the X and b r.+s 3 k is in any positive root subgroup then [a,b] can be ignored. Similarly we can rearrange the desired terms using ab = ba[a,b]. NOw denoting all irrelevant terms by y we have in Cases I, II, and III Om-l [xs(u).xR(t)] = [xso(u).xro(t) °°°xro(t) ] om—l = [x (u).x (t) 1 °°°[X (u).x (t)] 8o ' r0 30 r0 Y [ t ) (t)] ( ) < )“m-ll = x .u ,x ... u 'x t so rO [x30 r0 Y [ ( )t < )1 [ ) < )IGm-l = x u ,x t x (u ,x t 80 r0 80 re Y ' om-l = xro+SO(NSOaI-'Otu) . o o xro+so(Nso'rOtU) y = XR+S(NsO,rOtu)Y" where N = i1 is a structure constant of the Lie 8o'ro algebra of type Q over C. The computation for Cases IV, V, and VI is similar. -+s and r 4-s are not roots we have Noting that r 1 1 O O (u)xs (u)o.xr (t)xr (t)O]y [X (u).x (t)] = [X s R o o o o = [xso(u)'xro(t)][xso(u)'xro(t)]cy N tu)y x ( R+S so,rO 47 Thus we have Lemma 3.5.2: If R,S are adjacent fundamental roots in the Dynkin diagram of 1MK) then [xs(u),xR(t)] = xR+S(itu)y for t 6 LR' u 6 LS , where y consists of terms involving roots in Q of height greater than 2. The sign is independent of the field K. we will need to consider roots R.S as in diagrams II and III in greater detail. For R and s as in diagram II the positive roots of 9 ‘which lie in the span of R U S are the roots r . r1. s , r 4-s O 0 O 0' 4-s0, and roi-r14-so. The equivalence classes for I:1 these roots are 3.5.3 R = [ro.rl} S = {50} R +S = {ro+so,rl+so} 2R +S = {ro+rl+so} If |K1 = q, the elements of the root subgroups are _ 9 2 3.5.4 XR(t) - Xr (t)xr (t ) t E GF(q ) O l xs(t) = xso(t) t E GF(q) XR+S(t) = xr +s (t)xr +s (t9) t E GF(qz) O O l O x2R+S(t) = x (t) t 6 (EM) . 48 By using the Chevalley commutator relations one can show that 3.5.5 [xs(u),xR(t)] xR+S(itu)x2R+S(itq+lu) (t(tu4-tquq)) [XR+S(u)'xR(t)] x2R+S [X2R+S'XR] = [XR+S'XS] = [x2R+S'xS] [x2R+S'XR+S] = 1 ' where the signs are independent of K, and t and u range over the appropriate field in 3.5.4. For 3D4 with fundamental roots R,S as in diagram III, the positive roots are r0. r1. r2, so, ro+so, r1+80, r2+80, r1+ 1:24.80, ro+r2+so, ro+r1+s r0+r1 24-30 and r0+r1 equivalence classes 0' +-r +-r24-Zso. These roots form 3.5.6 R {ro.r1.r2} S = {so} R+ S = {r0+ so,rl+ sent2 + so} 2R+S = {r1+r2+so,ro+r2+so.r0+r1+so} 3R+S = [ro+r1+r2+so] 3R+ 25 = [r0+rl+r2+250} With |K| = q, the elements of the root subgroups are 49 3.5.7 2 me = xr (t)xr (tqmr (tq ) teGF.) = XR,1(t) . Acting on both sides of this equation with ”l we have 58 q"4 x 2(fR(t>.)) xR'1(tx) RP O -1 h c"4, = (xR 1(t) ) ¢ (h 4)‘1 v (xR'1(t) 4) = XRP 2(fR(t)kR(A)) o I by 4.1.6. Thus for t 6 LR 1 and l 6 HR 1, fR(tk) = fR(t)kR(l). Setting t = 1 yields fR(x) a fR(l)kR(X); 'we now have 4.1.7. fR(tx)f(1) = fR(t)fR(x) for t e LR'l. 1 e HR.1 . With the following two lemmas we remove the restriction on X in 4.1.7, with one possible exception. Lemma 4.1.8: Let L1' L2 be locally finite fields and let H be a subgroup of L: such that no prOper subfield of L1 contains H. Let f :L1 4 L2 satisfy the conditions (i) f(tl+-t2) = f(t1)+-f(t2) for t1. t2 6 L1: (ii) f(1112)f(l) = f(11)f(12) for 11. l2 61H. Then f satisfies (ii) for all Al. 12 E L Further, 1. either f is identically O on L1. or else the function u(t) = f(t)/f(l) is a field homomorphism. 59 ‘ggggg: We first show that f satisfies (ii) for all ll. 12 6 L1. Now, the additive closure of H in L1 is closed under multiplication, and also inverses since L1 is locally finite. By the hypothesis on H, L1 is the additive closure of H. Thus for s,t 6 L1 there are elements *1"°°"m' T1.....T 6 H with s = Zij and n t a ZTk' Expanding and rearranging f((ij) ( ZTkHfll) . using (1). (ii), and the distributive law, one easily shows that f(st)f(l) = f(s)f(t). Further. if f(1) = 0 then for any t 6 L1 ‘we have f(t)2 = f(t2)f(l) = 0, so f(t) = o. On the other hand, if £11) #'0 then the function u is defined. Clearly, u preserves addition and multiplication, and u(l) = 1 so u is a field homomorphism, completing the proof of Lemma 4.1.8. Lemma 4.1.9: Let L be a finite field and let H be a subgroup of L* ‘with |L| > (|L* :H1-1)2. Then no proper subfield of L contains H. Proof: Let L0 be the additive closure of H in L. Then L0 is a subfield of L. Let q = |L|, let d be * * * the degree of L over L and let m = IL :LOI g_|L :H‘. 0' Then it suffices to show that q_O we must have q gm-l. Thus qg(m-l)2 as desired. New suppose that 1{>(Kl) #'2A2(K1). Then by * Lemma 3.4.8. lLR 1 :HR 1| g_2. so by Lemma 4.1.9, no prOper subfield of LR,1 contains HR,1 whenever IKl‘ > 1. On the other hand, if 19(K1) = 2A2(Kl) then IL; 1 :Hh 1| $.3: thus, by Lemma 4.1.9. no prOper subfield of LR.1 contains H 2_ 2 R,1 ‘whenever [K1| - ‘LR,1‘ > 2 . Thus we conclude 4.1.10. If 1§(K1) #’2A2(2) then for R e 23. no prOper subfield of LR 1 contains HR 1' From 4.1.4. 4.1.7. 4.1.10 and Lemma 4.1.8 we obtain 1 2 4.1.11. If §(K1) #' A2(2) then for all t1. t2 6 LR,1' fR(t1t2)fR(l) = fR(tl)fR(t2) . We obtain a second consequence of 4.1.10 as follows. Regard LR,l and LRp 2 as subfields of E%, and let _* L0 = LR,1 n LRP,2. Now, Kp is locally cyclic and so contains at most one subgroup of a given finite order. As the map kR in 4.1.6 is faithful, we have HR,1 S-HR,2’ so HR,l g LO. By 4.1.10, either L0 = LR,l' which implies 61 i _ 2 . that LR.l g.L . or 5(K1) - A2(2). Even in the RP,2 _ 2 - latter case we have LR,1 — GF(2 ) g_L p since L P R '2 R '2 is an extension of degree 2 over K2. Thus we have 4.1.12. LR,1 S LRp,2 for all R 6 Z). In Section 4.2 we let m2 = elx‘l, where x is a certain diagonal automorphism of iNKZ). In Section 4.3 and 4.4 we let $3 = ezg'l and $4 = $37-1. respectively. where g is a graph automorphism and T is a field auto- morphism of i“1(2). In each section. the function fR is redefined to satisfy a version of 4;l;§_where the apprOpriate ”j replaces m1. We nose that 4.1.11 :nd 4.1.12 depend only on the fact that Bl1 g.B2 and N11 g N2. Since diagonal, graph, and field automorphisms of i§(K2) leave B2 and N2 invariant, 4;l;§, 4;1;ll and g;l;lg_hold for the redefined functions fR. As in the Steinberg argument on the automorphisms of Chevalley groups, the diagonal automorphism is used to obtain fR(l) = l for R e A and the graph automorphism is used to eliminate the permutation p in 4;l:2, while maintaining the identity fR(l) = l for R e A. Then fR is a field homomorphism for R E A. After applying the field auto- morphism T, we obtain fR(t) = t for R E A and t e L R.l The remainder of this chapter is devoted to showing that $4 is the identity map on l@(Kl). 62 ’ In view of the hypotheses of Theorem.C. we ignore the exceptional case l“1(1) = 2A2(2) when it arises in the remainder of this chapter. 4.2. Diagonal Automorphisms. By 3.4.5 there is a diagonal automorphism x of l§(K2) such that x (t)x = x (f (l)t) for R e A. t e . RP,2 RP,2 R LR'Z Thus if we let $2 = wleI ‘we have q, . “1mm 2 -- xRp 2(fRo_o R S = o——o==o——o A graph automorphism of 2E6 must therefore permute the two roots R and S in 4.3.2. Since the roots of E6 which are contained in R U S span a root system of type 64 A3 we may apply the argument for 2A3 to show that p must be trivial. Finally. for 3D the set of positive roots is 4' described in 3.5.6. 3.5.7 and 3.5.8. The first and fourth commutators in 3.5.8 are clearly not identically 1. To see that this is true for the second, third and fifth commutators as well. one needs to show that the equation 2 xq 4-xq4-x = O is not an identity in LR = GF(qB). If q is not a power of 3 then for t e GF(q)\O. tqzi-tq4-t = 3t y'o. If q is a power of 3 then for a solution t of xq a x4-l we have t(13 = t so t 6 LR and one can check that (t2)q2+(t2)q+t2 = 2. Thus none of the first five commutators in 3.5.8 is identically 1. For this reason there are three non-fundamental roots . + P in 23 such that [XR,1'XP.l] #’1. thus there are at least three non-fundamental roots Q in 2+ such that [ ,XQ 2] #'1. Since Rp is either R or S. X RP,2 Xs 2 commutes with all X 3R4-S, ‘we must have Rp = R. Thus p is trivial, and and Q 2 for Q in 23+ except this completes the proof of 4.3.1. Now, let $3 = szg‘l 'with g as in 4.3.1. Then we have q’3 4.3.3. xR’lgch'2 for R 622. we again redefine fR to satisfy 65 ”3 4.3.4. (t) XR,1 e XR,2(fR(t))X2R,2 for R.e Z). t e LR,l’ For the untwisted Chevalley groups the automorphisms g (201 for groups of type A‘, D‘ or E6, 301 for groups 2 of type D4, 02 for groups of type 82 or F 2 4. and 03 for groups of type 62) all have the prOperty that (l) = x (l)9 for R e A . XR'Z Rp 2 0 Thus fR(l) = l for R 6 A. so 4.2.2 still holds. For other roots, 4.1.11 holds. Further, 4.1.12 now becomes 4.3.5. LR,1 S-LR,2 for R e Z). Also note the easy consequence of 4.3.5: 4.3.6. K1 3 K2 4.4. Field automorphisms. Let L1. L2 be the extensions of K1, K as described in Section 3.1 and 2 listed in Table 3. Then for R E Z) and j = 1,2. KjSLR'j g Lj' we now show that 4.4.1. There is a field homomorphism f :L1 4 L2 such that f |LR 1 = f for all R e A. R 66 Keep in mind that at this stage the fR satisfy 4.3.4 and 4.2.2. To verify 4.4.1 let R,S E A be adjacent in the Dynkin diagram of ZL It follows easily from 3.5.1 and ALA. that fR+S(t1t2) = fR(tl)fS(t2) f°r t1 6 LR.1' t2 6 Ls.1 Thus by 4.2.2 4.4.2. For t 6 LR 1 n L8 1. fR(t) = f R+s(t'1) = fR+S(l-t) = fS(t) . Now. for untWisted groups, Kl = LR.1 = LS,1 = L1. Thus fR = fS for adjacent R, S e A. Then the connectedness of the Dynkin diagram ensures that fR = fS = f for all R, S e A, verifying 4.4.1 for these groups. For the twisted groups, let A = [R .,Rm} ‘with the 1". R. labelled as in Table 3. Note that L = LR and that J l 1,1 .2 ‘2 ... 2 . Let f = f . It follows LRlll LRzll I‘Rm'l R1 easily from 4.4.2 that f ‘LR l = fR for j = l....,m. j' j completing the verification of 4.4.1. Now f(t) = tq for t 6 L1, where q is some power of p. The automorphism ¢ = oq of §(K§) commutes with the automorphism loq where q2 = |K2|, so T 2 67 restricts to an automorphism of 1@(Kz). Letting $4 = @37-1 it is easy to see that we have m 4 4.4.4. XR.l(t) e xR'2(t)x2R'2 for R e A, t e LR’l In the next three sections we use 4.4.4 to deduce that q4 is trivial on 1§(K1). In section 4.5 we consider the Chevalley groups in which the type of each MR/C(HR) is Al, showing that ”4 .ésélil. x = x for x e XR. R e A . In 4.6 we verify 4.4§*2 for the groups in which the type of .NR/Q(MR) is 2A2. 2B2 or 262 for some R E A. Finally in 4.7 we show 4.4g*z e o4 |14(K1) = 1 . which shows that c :1§(K1) 4 19(K2) extends to the automorphism Tng of lMKZ). 2 2 3 2 4.5. Gro s of t s A . . D , and E _. _JL—m. “-1 DI 4 __ 5 and all untwisted groups. For these groups each MR/C(MR) for R E Z is of type A Thus for each R e Z, 10 X2R = 1. Then 4.4.1 becomes o 4.5.1. xR,l(t) 4 = XR,2(t)' for R e A, t e LR.l' 68 To obtain 4.4§*2 we therefore need only show that for R e A and t e LR,1 ‘we have xR,l(t) = xR'2(t). But this is just the conclusion of Lemma 3.3.4. In view of 4.3.6 'we thus need only show that 4.5§*) The degree of the extension K2 over K1 is relatively prime to i. For the untwisted groups i = l and 4°5§*2 is clear. In each of the root systems of type 2A2‘_1. 2 1' and 2E6 there are roots, R,S E A which span a root system of type 2A3. as described in Section 3.5. By D 3.5.5 we'have [xs(u).xR(t)] = xR+s(ttu)x2R+s(ttq+lu) By 4.4.4 we have ”4 [xs'1(u).xR'l(t)] q1+l = “Ra-s,2(*fR+s(t“))"2R+s.1(*sz+s(t u” ' Computing another way and using 4.5.1 we have ”4 ( )”4 ”4 [x8 1(u).xR,1(t)] [xS 1.u .xR l(t) ] [xS 2(U):XR'1(t)1 q2+1 = "R+s.2("="'“)"2R+s,2”‘t “) 69 q1+1 q2+1 2 Thus f2R+S(t u) - t u for all t E LR,1 - GF(ql) and all u 6 L5 1 = GF(ql). Setting t = l ‘we have f2R+S(u) = u for t e GF(ql) a L2R+S,1’ hence f2R+S is the identity map. Setting u 1 we thus have q +1 q +1 t 1 = t 2 for t G GF(qi) . Now choose t e GF(qi) to have multiplicative order q1+-l. q2+1 Then t 1 so q14-1 divides q24-l. Let m be over K . Then the degree of K2 1 q14-l divides qT4-l. But ql e -1 mod q1+l so qT-i-l a (-l)m+l mod ql+l. Thus we must have m odd. As i = 2 for the groups 2 2 . . 2‘_]_. 2DL and E6. we have verified 4.5§*2 for these groups. A For the groups of type 3D4, the root subgroups XR for R 6 23+ are described in Section 3.5. By 3.5.8 we have 2 [Xs(u).xR(t)] = xR+S(*tu)x2R+s(*tq +qu) . 2 q +q&l 2 (*tq +q+lu2) (it u)x x3R+s 3R+ zs and by an argument essentially identical to the argument above we obtain 2 2 ql+ql+l q2+q2+l 3 t = t for t e GF(ql) 70 Here we choose an element t e GF(qi) which has multiplicative order qia-q14-l. As before we obtain 2 . . 2 q1 + q1. + l diVides q2 + q2 + l . Multiplying by q1-l, and letting m be the degree of K over K we find that qi-l divides (qim+qT+ l)(q1-1). 2 1' New suppose m is divisible by 3. Then qim a q? E 1 mod qi-—1 so qi-—l divides 3(q1-1). This implies that qi4-ql4-l divides 3 ‘which is impossible for q1 > 1. This contradiction shows that m is not 3 divisible by 3. As i = 3 for D4, we have shown 4.5(*) for this case. Therefore we have obtained 4.5§*2 for all of the groups in which each subgroup MR has type A1. and consequently we have 4.4§*2 for these groups. 2 2 2 2 4.6. Groups of types A22. B2. F4, and 62. The first step towards proving 4.4(*2 for these groups is to show that 4.Sj*2 holds. Now, the groups 2B2(2m). 2F4(2m) and 2G2(3m) are defined only for odd m. Since i = 2 in each case. this ensures that 4.5(*) holds for these groups. For the 2 groups of type A2L' let R be the root labelled R1 . 2 in Table 3. Then MR/Z(MR) has type A2. Then by 4.4.4 094 . xR l(t) = xR 2(t)z(t) for t 6 LR l'Wlth z(t) 6 X2R 2 71 By 3.2.2.10. X so for t 6 L 2R,2 = C”‘21:.” R.1 w [xR'luth'lun 4 = [xR'2(t)z' By 3.2.2.1, 3.2.2.6, and 3.2.2.7, Y = . ro+r1 -ro-r1 It is easy to show from Lemma 3.1.1 that YX/Z(YX) a A1(K). For 262(K), ‘we consider the subgroup Y = (X2R(S).x_2R(t) ‘s,t E K). By 3.2.4.1 we have 6 6 Y1= of G2(K). Since neither the sum nor the difference of r+r and r+3r O 1 O Chevalley commutator relations [3, Thm. 5.2.2] that 1 are roots, it follows from the (M. ,M > = M )(M . By Lemma 3.1.1, r0+r1 rd+3r1 ro+r1 r0+3r1 ——- 2 M /§(M ) a M /C(M )2 A (K). Now G (K) ro+r1 ro+r1 ro+3r1 r0+3r1 1 2 is the subgroup of G2(K) generated by the unipotent elements fixed by a certain automorphism, 0, of order 2. As o interchanges M and M , ro+rl ro+3r1 of E generated by unipotent elements of E which are Y, the subgroup fixed by 0, is such that Y/C(Y) a A1(K). ‘We conclude that either G a PSL2(K) or G a Sz(K) for some infinite locally finite field K. To complete the proof of Theorem D'we note that if K1 is an infinite proper subfield of K then G contains either PSL2(K1) 84 or Sz(Kl) as an infinite simple prOper subgroup. 5.2. Locally finite fields. We now obtain a classification of the locally finite fields, with the objective of characterizing the fields K which appear in Theorem D. Suppose K is a locally finite field; that is, every finite subset of K is contained in a finite sub- field of K. Obviously, K must have characteristic p for some prime p. Now, every element of K has finite multiplicative order so the elements of K are algebraic over the prime subfield GF(p). Thus K is contained in the algebraic closure ‘K% of GF(p). Conversely, any subfield of 'Kfi is locally finite. To see this, let X be a finite subset of ‘K%. Each element x e X is a root of some polynomial fx ‘with coefficients in GF(p). Then X is contained in the extension F of GF(p) generated by roots of the polynomial f = U f . The degree of F over GF(p) is less than xex X or equal to the degree of f, so P is finite. Thus the classification of locally finite fields reduces to the classification of the subfields of the fields §%. Let Efp= [K|Kgip} . 85 where the symbol "g” means "is a subfield of". Also let GFp = {F 3 Tip | F is finite} Now, the fields in GFp are easily classified. If F 6 GFP has order pn then F2 GF(pn). Further, there is precisely one subfield of ‘Ki which is isomorphic to GF(pn); namely the subfield of 'K n roots of the polynomial f(x) = xp -x. consisting of Thus there is a one-one corresPondence between the elements of GFP and the set 11 of positive integers, whereby the subfield GF(pn) e GFp corresponds to the element n e N. Moreover, both GFp and N have lattice structures with meet, A, and join, V, defined by 5.2.1. For F1. F2 6 GFp, F1 A F2 = F1 n F2 and F1 V F2 is the subfield of 'Ké generated by F1 and F2. 5.2.2. For n1, n2 6 N, n1 A n2 is the greatest common divisor of n1 and n2, and 111 V n2 is the least common multiple of I11 and n2. It is easy to verify that n n n An GF(p 1) A GF(p 2) = GF(p l 2) , and n 112 n1Vn2 GF(P 1) v GF(p ) ll 6) '11 'U 86 5.2.3. The correSpondence n a GF(pn) is a lattice isomorphism between GI"p and N . Now, the lattice Operations defined in'§;g;l extend to lattice Operations on 'Gffi. With respect to these lattice operations, '55? is complete; that is, every subset of 55% has both a greatest lower bound and a least upper bound. Further, it is clear from the definition of local finiteness that each field K 6'55? is the union of the finite subfields of K, so for K e 55?, 5.2.4. K V{F |F 6 3K}, where 3K {FEGFPIFgK} . Thus we may regard '55? as the completion of the lattice GF . P The subsets 3 = 3K E G1='p satisfy 5.2.5. (i) If F V F E 3 ; and ,F2 6 3 then F1 2 1 (ii) If Fl g_F2 and F2 6 3, then F1 6 3. Further, if ”J E GFp satisfies 5.2.5 then by 5.2.5 (i), the union of the fields in 3 is a subfield K of 'Kfi, and then 5.2.5(ii) ensures that 3 = T Moreover, it K. is easily seen that for K1, K2 6 GFP, 3K1 = 3 2 if and only if K1 = K2. Thus 87 5.2.6. The correspondence K a 3K between 5?? and the collection of subsets of GFP satisfying 5.2.5 is one-one. Now, for K 6 55% let 5.2.7. .BK = {n e N |GF(pn) 3 K} . Also, we consider subsets .29 _c_ N satisfying 5.2.8. (i) If n1, n2 6 9' then nl V n2 6 fi‘: and (ii) If n1 divides 112 and n2 6 3. then u1 6 3» Clearly, the isomorphism in gng; establishes a one-one correspondence between subsets 3 E GFp satisfying _5_._2_£ and subsets .3 g N satisfying M in which 3K corresponds to 3K for K e 55?. To obtain a classification of the fields in ‘55? ‘we define a completion. ifi of II 'which satisfies a property analogous to 5.2.4. Let rl.r2,... be an enumeration of the set of primes. ‘We let i§ be the set of all formal products ‘where the exponents ej are either non-negative integers or the symbol a. We make no restriction on the number of non-zero exponents in v. For two elements d d2 e1 e2 __ u=r1 r2 ---,v=r1 r2 of N we define lattice Operations by ..., and min d ,e min d ,e 5.2.9. u A v = r1 { 1 13 r2 { 2 2} rTax{d1,el} rmax£d2,e2} t < < II In the obvious way, we may regard N as a subset of i; then the definitions 5.2.9 and 5.2.2 are consistent on IL For H: v e N we say p, divides v. written p, | v if uAv=u. For veN let 39 = [n e 1: In Iv} . It is easily seen that fly satisfies 5.2.8, and that for any subset .3 g N satisfying 5.2.8 we have .3 = 3v, where 5.2.10. v = A {n |n e 33 . Further, for v , v E i§, fi' = 3 if and only if 1 2 v1 v2 v1 = v2 SO‘We have 5.2.11. The correspondence v ”‘flv between EE and the collection of subsets of I! satisfying 5.2.8 is one-one. Finally, composing the correspondences in 5.2.6, 5.2.3, and 5.2.11 we obtain a one-one correspondence between 89, ‘GFE and, 35. In this correspondence an element v E i§ corresponds to a "generalized Galois field" GF(pV), where we define 5.2.12. GF(pV) = u {GF(pn) In 6 IN. nlv} . It is easily seen that GF(pv) is finite if and only if v E N . Thus we have shown Pr0position 5.2.13: Every locally finite field of characteristic p is a field GF(pv) for some v 6 N . __ v v For v1. v2 6 ll, GF(p 1) = GF(p 2) if and only if v1 = v2. For v 6 i§, GF(pV) is finite if and only if v E 11. Now, from PrOQosition 5.2.13 it is easy to show that an infinite locally finite field which contains no infinite prOper subfields must be one of the fields GF(pra) where p and r are primes. Using this fact, and taking into consideration the restrictions on fields K for which Sz(K) is defined, we obtain the following variation of Theorem D. Theorem D’: Let G be an infinite locally finite simple group in which every proper subgroup is solvable- by-finite. Then either G a PSL(K) ‘where K = GF(prm) for some primes p and r, or else G a Sz(K) *where K = GF(2r ) where r is an odd prime. We note that the groups listed in Theorem D’ have been studied in [6]. There it is shown that every infinite 9O subgroup of PSL2(K) is solvable with derived length at most 2, and every infinite subgroup of Sz(K) is solvable with derived length at most 3, ‘where K is as in Theorem D’. VI. CONCLUSION ‘§;l. Previous results towards Theorem B. Recently, a number of peOple have worked towards a proof of Theorem B. The first major result appears in [5, Thm. 4.18] where Theorem B is proved for Chevalley groups of types A 2A2. amd 2B Theorem B for all of the untwisted Chevalley types. In a ‘0 2. More recently Thomas [81 has proved- letter to Professor R. Phillips, Thomas mentions that he 2 2 2 has completed the proof for types A2n+l' Dn' E6' and 3D4, and is working on the remaining cases. Although the arguments in [5], [8], and our own work are similar in some respects, there are substantial differences in technique. §;3, Agconjecture.gg regular embeddings. Theorem C has potential importance for the theory of finite groups. However, to fully exploit this potential, one needs criteria regarding an embedding‘ m :61 a G2, where G1 and G2 are Chevalley groups of the same type and characteristic, which will ensure that e is a regular embedding. Of course, the best we could hope for is the following 91 92 6.2.1. Conjecture: Every embedding of one finite Chevalley group into another finite Chevalley group of the same type and characteristic is a regular embedding. Suppose that Gl g_G2, where G1, G2 are Chevalley groups of the same type and characteristic. To prove 6.2.1 we must do two things. First we must show 6.2.2. There is an element 9 6 G2 such that Bi 3 32. Having shown 6.2.2 we may then suppose without loss of generality that Bl g_B2. Then we must show 6.2.3. If BISB2 then there is an element b E B 2 such that 1011’ 3 N2. Now, the proof of 6.2.2 can be obtained as in the proof of 2.2.2 if it can be shown that 6.2.4. Each Sylow p-subgroup of G1 is contained in a unique Sylow p-subgroup of G2, ‘where p is the characteristic of G1 and G2. ‘We showed in the proof of'g;2;2_that gég;g holds cofinally in any ascending chain of Chevalley groups of fixed type and characteristic. In a conversation with B. Hartley, he indicated that §&g;4 can be proved, with certain exceptions involving rank one Chevalley groups, 93 by showing that the nilpotence class of any intersection of distinct Sylow p-subgroups of G is less than the 2 nilpotence class of a Sylow p-subgroup of 61' For the rank one Chevalley groups, 6.2.4 follows from the fact that distinct Sylow p-subgroups in a rank one Chevalley group must have trivial intersection. This is shown in PrOposition 6.2.5 below. The proof of 6.2.3 may be more troublesome. One possible method of proof is suggested in the proof of 2.2.4. The success of this method depends only on being able to show that each Hall p‘-subgroup of B is contained in 1 a unique Hall p’—subgroup of B The proof of 2.2.3 2. (given at the end of Section 2.2) suggests that this is often the case. However, there are certain exceptions, as in embeddings of A1(2) into A1(2n) where we have HI = 1. we conclude the evidence for 6.2.1 with a prOposition which suggests an alternative method for proving 6.2.3. Proposition 6.2.5: Let G1, G2 be Chevalley groups of rank one and characteristic p, with G1 g.G. Then the embedding of G1 into G2 is regular. Proof: In a rank one Chevalley group G the weyl group N/H has order 2, and for any n e N\H we have G=BUBnB. 94 Thus if g 6 G and U9 #’U, then there are elements b’, b e B such that g = b' n b. Then I unU9=UnUbnb=(Unfl)b=(unV)b=1. That is, distinct Sylow p-subgroups of G intersect trivially, and 6.2.4 follows easily. As noted above, we may thus suppose that B1‘g_B2, and the proof of PrOposition 6.2.5 will follow if we verify 6.2.3. We do this indirectly, obtaining an element b E B such that V? g.V2, then showing that N? g'Nz. 2 Since V1 is a p-subgroup of G2. ‘We have V? g,U2 for some x 6 G Clearly U2 cannot contain both U1 2. and V1 so it follows that x f B2. Thus there are elements b, b E B and n 6 N2\H2 such that x = b n b . 1 2 l ln-l -1 Then Vinbl g_U2, so V? g.UgI = Un = V 2 2' Conjugating by b, ‘we may suppose that Ul S.U2 and V1 g V2, and we must show that Nllg.N2. we also have Ulfil = B1 S_B2 = U Vlnl-S'VZHZ' Then 2H2 , and similarly obtain H =UH nleIgUH RVZH =H l 1 1 2 2 2 2 ' Thus, to show that N1 g N2, it suffices to show that for n1 6 N1\H1 and n2 6 N2\H2 we have n1H2 = n2H2' This is verified easily by showing that nln'z'l nOrmalizes both U and V 2 2, as in the proof of Lemma 3.1.6. BIBLIOGRAPHY BIBLIOGRAPHY M. Aschbacher, A characterization of Chevalley groups over fields of odd characteristic, Ann. of Math. (2) 106 (1977). 353-398. N. Burgoyne, R. Griess and R. Lyons, Maximal subgroups and automorphisms of Chevalley groups, Pac. J. of Math., 71 (1977), 365-403. R.w. Carter, Simple Groups of Lie Type, J. Wiley and Sons, N.Y. (1972). K.A. Hirsch, Periodic linear groups, Proceedings of the International Conference on the theory of groups held at the Australian National University, Canberra, 10-20 August, 1965, Gordon and Breach, London (1967). O. Kegel and B.A.F. Wehrfritz, Locally Finite Groups, NOrth Holland, Amsterdam (1973). A.A. Shafiro, Examples of locally finite groups, Mat. Zametki, 13 (1973), 103-106. B.A.F. Wehrfitz, Infinite Linear Groups, Queen Mary College Mathematical Notes, London (1969). S. Thomas, An identification theorem for the locally finite nontwisted Chevalley groups, Arch. Math. (to appear). 95 R M'CJITI'I‘ItflflwijJMfiIflfljllzflfluiflflflfliwWI“