HI WWWWIHIHHUIHMH ! WWWWWI/NH IHESJS 31293 01691 13 This is to certify that the dissertation entitled Reswiucd pfmper’h-icg cg Fin'dtav‘s LUIWQ‘V Groups presented by [\Q lei/“(AA RC‘€\£DVC\ has been accepted towards fulfillment of the requirements for “9%: Dc degreein ”I? t/7f‘9/7197/C5 WM” 57/4254; Major professor Date AMJ! :8) W171] MSU is an Affirmative Action/Equal Opportunity Institution 0-12771 LIBRARY Michigan State Unlverslty PLACE IN RETURN BOX to remove this checkout from your record. or before date due. TO AVOID FINES return on DATE DUE DATE DUE DATE DUE /..——— ______._,._.—— _________._——— l // ___,_/ xf / / / /———- /.——— / l // / // 1/98 c'JCIRCIDateDuo.p65-p.14 RESIDUAL PROPERTIES OF FINITARY LINEAR GROUPS By Aflahiah Radford A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1997 ABSTRACT RESIDUAL PROPERTIES OF FINITARY LINEAR GROUPS By Aflahiah Radford In this thesis, we generalize a theorem of Bruno-Phillips describing certain locally finite finitary groups (modulo its unipotent subgroup) as a subdirect product of fi- nite dimensional groups. Our only restriction is that such groups are residually finite and this forces all the composition factors of the module to be finite dimensional. Specifically we have Theorem 4 which states: Let G S FGL(V, K) be such that G is both locally finite and residually finite. Then every G—composition factor of V is finite dimensional and G / unip(G) is a subdirect product of finite dimensional groups. We showed that any irreducible finitary linear group has the conjugate centralizer property with respect to any of its normal irreducible subgroup. In the case of prim- itive linear groups of infinite dimension, all normal subgroups are irreducible and if such groups are also locally finite then we find that their derived groups are simple (Theorem 3). Theorem 3 is used in the proof of Theorem 4 and the proof of Theorem 3 makes used of J. Hall’s classification of simple, locally finite finitary linear groups. This allows us to describe certain subgroups as a direct product of two entities; a direct product of simple groups and subdirect product imprimitive groups, a crucial step in proving our main result. ACKNOWLEDGMENTS I would like to thank Dr. Richard Phillips for all his help, encouragement, advice and patience throughout my graduate work at Michigan State University. He is an excellent adviser. His knowledge of Group Theory is only surpassed by his kind and generous personality. I would also like to thank my committee members, Dr. Jonathan Hall, Dr. Ulrich Meierfrankenfeld, Dr. Susan Schuur and Dr. Edgar Palmer for reviewing my thesis and for all their suggestions, comments and help they gave me. iii To my husband, Glenn Harley Radford iv TABLE OF CONTENTS Introduction 1 Chapter 0 5 1 Conjugate Centralizer Property of Imprimitive Groups and Stable Groups 8 2 Simplicity of the Derived Groups of Primitive Groups 30 3 Composition Factors of Residually Finite Groups 40 BIBLIOGRAPHY 46 Introduction Let V be a vector space over a field K. An element 9 6 GL(V, K) is finitary if [V, g] = {v(g— 1) | v e V} has finite dimension. The degree of g, denoted by deg(g), is the dimension of [V, g]. Observe that g is finitary ifi 0,,(g) = {v E V I 229 = v} has finite co-dimension in V. The group of finitary transformations of V, denoted by FGL(V, K), is de- fined as the set {9 E GL(V, K) | g is finitary}. The group FGL(V, K) is a normal subgroup of GL(V, K) and seemed to have been first introduced by Dieudonne’ [2]. Rosenberg [14] used the concept of finitary linear transformation as an essential fea- ture in classifying normal subgroups of general linear group. A K-finitary representation of a group G is a homomorphism 2/) : G —) FGL(V, K), from G into FGL(V, K) for some K-vector space V. The representa- tion is faithful if ken/J = 1. Groups with faithful K~finitary representations are called K-finitary linear groups and if the associated vector space V is finite dimensional, then G is called finite dimensional . Some examples of finitary groups are: 1. subgroups of GL(n,K). 2. groups of finitary permutations 3. GL(n, K )wm(G, Q) where (G, 52) is a group of finitary permutations 4. Stable groups. 2 Obviously subgroups of GL(n,K) are finite dimensional groups. If (2 is a non-empty set, then 9 E Symm) is a finitary permutation if supp(g) = {a 6 Q | 0:9 919 a} is a finite set. The set {9 E Symm) I g is finitary} is denoted by Sym(Q,No) and finitary permutation groups consist of all subgroups of Symm, No). For any K, the finitary permutation groups (G, (2) acts faithfully on the K-space V = K 9, the permutation module of G with basis {220 | a E Q} by 223, = 1109 . Results from the theory of finitary permutaton groups are used often in this thesis. If V is a K-vector space, E a basis of V, then let stab(V, E‘.) = {g 6 GL(V, K) | v9 = v for all but a finite number of v E 3}. Hg is another basis, then stab(V, E, K) is conjugate to stab(V, 8, K) in GL(V, K). This common isomor- phism type will be called the stable general linear group and denoted by Stab(GL(V, K)). Subgroups of Stab(GL(V, K)) are called stable groups. Obviously stable groups are finitary linear groups. Stable groups play an important role in the sequel. We prove in Chapter 1 that if G S FGL(V, K) is an irreducible finitary linear group with dim(V) being countably infinite, then G has a stable basis. Thus G is a stable group. This fact is used in the proof of one of the main results of this thesis: Theorem 2: Let G S FGL(V, K) be an irreducible, primitive and locally finite finitary group where V has infinite dimension. Then G’ is simple. In order to prove our result above, we made used of the following classification of locally finite, simple groups of FGL(V, K) by J. Hall [4] which states: Let G be a locally finite, infinite, non- “finite dimensional ”, simple subgroup of FGL(V, K) , where dimension V is infinite. Then G is either alternating, the derived subgroup of FSp(V, K, a), FU(V, K, h) or FO(V, K, q), or a group of type T(W, V), W Q V“ and annv(W) = 0. 3 Here a, h, q are respectively non-degenerate alternating, Hermitian and quadratic forms on V. The special transvection group T(W, V) = < t(oz,a:) | a E W, x E ker(a) > where W g V" and the map t(oz,a:) E GL(V, K) is defined by t(a,a:)(u) = v + ((u)a)a: where u E V. In general T(W, V) is not simple but a sufficient condition for simplicity is annV(W) = {u 6 V | (u)a = O for all a E W } = 0. In the proof of Theorem 2 we show that G’ is actually one of the simple groups listed above. The proof of Theorem 2 makes substantial use of the the conjugate centralizer property. A group X has the conjugate centralizer property relative to its subgroup H if for every finitely generated subgroup F of G, there'exist an element h 6 H such that [F, F"] = 1. Specifically, we show that (i) If G S FGL(V, K) is primitive, then G has a unique component M, and (ii) G has the conjugate centralizer property with respect to M. From these two results, together with the consequence of the conjugate centralizer property, we are able to show that M = G’ and G’ simple. It is in the proof of this that that we heavily use stable groups and the classification of J .Hall. In addition, we will establish Lemma 2: Let G S FGL(V, K) be an irreducible, infinite dimensional and imprimitive fini- tary linear group that is also locally finite. Let N be a normal irreducible subgroup of G. Then G has the conjugate centralizer property with respect to N. A primary consequence of this result is: Theorem 1: Let G g FGL(V, K) be an irreducible, infinite dimensional and imprimitive fini- tary linear group that is locally finite. Then G’ is the unique minimal subnormal irreducible subgroup of G. Our determination of the structure of primitive groups permits extension of results of Bruno and Phillips [1] regarding residually finite, finitary linear groups. In [1], it is shown that certain types of residually finite, finitary linear groups are subdirect products of finite dimensional groups. Here in Chapter 3, we are able to prove the following: Theorem 4: Let G S FGL(V, K) be such that G is both locally finite and residually finite. Then 1. every G-composition factor of V is finite dimensional. 2. G / unip(G) is a subdirect product of finite dimensional groups. Recall that a group X is residually finite if 1 = fl{H 31 X I X / H is finite}. This is equivalent to “if 1 91$ 1: E X, 3H 31 X such that IX/Hl < 00 and a: g H” . Chapter 0 In this chapter, we list some definitions and significant results that will be used but are not stated in the introduction nor in the following chapters. Definition 1 Let G g FGL(V, K) be a group and H a subgroup of G. Then the degree of H, denoted by deg(H) is the dimension of the vector space [V, H]. Definition 2 Let A Q FGL(V, K), a non-empty subset of FGL( V, K )and Y Q V , a non-empty subset of V. Then 1. = and 2. [Y,A]= . Note that < YA > and [Y, A] are < A >-subspaces of V. The following Lemma 1 from [15] is used in Chapter 1. [15, Lemma 1]. Let G: < T > Q FGL(V, K) where T is a finite set. Then 1. dim(V/CV(G)) is finite. 2. dim[V, G] g Z{dim[V, t]|t E T}; thus dim[V, G] is finite. 3. If W is finite dimensional subspace of V, then < WG > is finite dimensional. 4. There is a finite dimensional G subspace X of V such that dim(X) is finite, [V, G] g X and G acts faithfully on X; further there is a subspace Y of CV (G) such that V = X 69 Y. 6 Some significant results from finitary permutation groups and finitary linear groups used in this thesis will be discussed next. Definition 3 A set U = {Vili E I} of subspaces of V is a G-system of imprimi- tivity of V if 1. V = EB{V,'|i E I}, and 2. for each g E G andi E I, there is aj E I such that V," = Vj. We will need the following properties of irreducible finitary linear groups. [12, 2.2.3]. Suppose that G g FGL(V, K) and that U = {V,-|i E I} is a system of imprimitivity of G with |I| > 1 on which G acts transitively. Then 1. Ifi E I, then V,- is finite dimensional (and all of the V,- have the same dimension). 2. G7r is a group of finitary permutations. 3. ker(7r) is a subgroup of the direct product of groups L,- Q GL(V}, K) and each L,- is an image of ker(7r) ; note that the groups L, are finite dimensional K-linear groups of bounded degree. Thus by (2) above G / ker(1r) is an infinite transitive group of finitary permutations. This connects the irreducible finitary linear groups with the transitive permutation groups. We will next list some crucial properties of transitive finitary permutation groups. Let X be an infinite transitive group of finitary permutations on a set I). Since X is finitary, the X-blocks are finite sets. Furthermore either (I is the union of an ascending chain of a countable number of blocks (in which case X is called totally imprimitive), or Q has a maximal block (in which case X is called almost primi- tive). If X is totally imprimitive, then there is a chain N1 Q N2 Q .. . Q N,- Q . .. of normal subgroups of X such that 1. U{N,-} = X, and 2. each N,- is a subdirect power of a finite group. Therefore both I) and X are countable. On the other hand if X is an almost primitive group, then X has a normal subgroup N such that 1. N is a subdirect power of a finite group and 2. G / N is either the alternating group or the full group of finitary permutation on an infinite set. Definition 4 Suppose G g FGL(V, K), O = {V,-|i E I} a G-system'of imprimitivity and F a non-empty subset of G. Then the support of F, denoted by supp(F), is the set {Vi | i E I and Vif as V, for some 1’ E F}. Note that the support of F depends on a given system of imprimitivity. A result involving the support of F from [12] which is used in this thesis is the following: [12, Lemma 8]. Let G S FGL(V, K) and U = {Vili E I} be a system of imprimitivity of V with [I] > 1. Further, let A be a subgroup of G and let 3MF=WA13=UEHMAHWL For any element 9 E G, define 8(A)g = {j E J] for some i E S‘S(A),V§g = Vj}. Then 1. for all g E G, 3(A9) = S(A)g; 2- |3(A)| S |8UPP(A)| + deg(A); 3. if there is a g E G such that 99(A)g n %(A) = (D, then [V,A9] Q CV(A) and [MMECMMI Part (3) above implies [A,A9] = 1 (Lemma 9 of [12]). CHAPTER 1 Conjugate Centralizer Property of Imprimitive Groups and Stable Groups Definition 5 Let V be a vector space over the field K with basis X and G S FGL(V, K). Then X is a G-stable basis of V if each element of G moves only a finite number of elements of x. For countable irreducible groups G S FGL(V, K), we can always find a stable basis for G. This is the content of the following lemma. Lemma 1 Let G be a countable subgroup of FGL(V,K). Then V contains a G- subspace W, with the following properties: 1. G acts faithfully on W. 2. W has a G-stable basis x 2 {v0 | oz 2 1}. Proof: Enumerate the elements of G as {91, 92, g3, g4, . . .} and for each 11, let Gn be the group generated by {g1, g2, . . . , gn}. We then obtain an ascending sequence of finitely generated subgroups, G1 Q G2 Q Q G". Lemma 1 of [15] produces for each n, a finite dimensional subspace of V , X", such that Gn acts faithfully on Xn and [V,G,,] Q X”. Furthermore this lemma gives subspaces Yn with the property that V=Xn€BYn and K, Q CV(G,,). 9 Define SI = X1. For n 2 2, let 3,, = X1+ X2 + - - - + Xn and Jn be a direct compliment of Sn_1 n Yn_1 in Y -1. Claim 1 There exist disjoint sets {An}n21 E V such that (a) AIUA2°"UAk isabasisforX1+X2+~°+Xk VkZland (b) If1 g 2' < k then [Ah 0,] :0. Proof: Let A1 be a basis of X1 over K. Suppose V 1 < k < n, El disjoint sets A1, A2, . . . ,Ak E V such that 1. A1UA2"'UAk isabasisofX1+X2+---+X,c and 2. Ifl S i < k, then [Ak,G,-] = 0. Recall that Sn: X1 + X2 + - . - + Xn and Jn is a direct compliment of Sn_1 n Yn_1 in Yn_1 and Yn_1 E CV(G,,_1). Choose An to be a basis of 3,, fl Jn over K. Then V = Xn-1$ Yn—l = (X1+X2+~--+X,,_1)+Y,,_1 = Sn—l + Yn—l = 511—1 69 Jn (since Yn-l = (Sn—l fl Yn-l) EB Jn) = (X1+X2+"'+Xn_1)$Jn. Thus, Sn : X1+X2+'°'+Xn 10 = (X1+"'+X—1)$(Xnn']n) = (X1+...+Xn_,)@(s,,nJ,,) (since(X1+X2+°-'+Xn—1)an=0) = Sn_1 63 (Sn (1 J"). Therefore, since A1,A2, . . .,A,,_1 are disjoint sets and A1 U A2 U U An_1 is a basis for Sn_1 and A, is a basis of Sn n J“, A1,A2, . . . ,An are disjoint sets and A1UA2U”°UAnisabaSISOfSn =X1+X2+---+X,,. Observe next that for 1 g i < n, [Am G,] Q [5,, fl .1”, G,] = 0 since Jn g Yn—l g CV(Gn—l) and Gi g Gn—l- Cl Claim 2 [V, G] Quays... Proof: Since [V, Gn] Q X, Q 3,, and G = UnZIGn, [V, G] Q Un21[V, Gn] Q UnZISn. Cl Claim 3 The group G acts faithfully on Un_>_1 Sn. Proof: Suppose not. Then there exist k 2 1 and g1, g2 E G, such that gllun215n=g2|un213n. Thus the action of g1 on X k is the same as the action of g2 on X 1,. However G], acts faithfully on X k which leads to a contradiction. Cl Choose W = UnZISn. By Claim 2, W is a G-subspace and by Claim 3, G acts faithfully on W. By Claim 1, UnZIAn forms a basis of W over K and for each 11, Ga moves at most A1, A2, . . . ,An and fixes pointwise the set Uk>nAn. Furthermore the An’s are finite sets since the Sn’s are finite dimensional. Therefore UnZIAn forms a G-stable basis for W . 11 Note that if G g FGL(V, K) is irreducible, then W: V in the lemma above. The concept of the conjugate centralizer property was introduced by RM. Neu- mann in [8] as the tool for studying finitary permutation groups. It is also useful in working with finitary linear groups. Definition 6 A group G has the conjugate centralizer property relative to its subgroup H if for every finitely generated subgroup F of G , there exist an element h in H such that [F,Fh] = 1 The significance of the conjugate centralizer pr0perty is realised in the statement below. [1, 6.2]. Suppose that H g G and that G has the conjugate centralizer property relative to H. Then 1. G’ Q H and 2. ifT_<_lH,thenT§1G. In the case where G is locally finite, the subgroup F in the definition of the conjugate centralizer property above is a finite group. The ultimate goal here is to show that any irreducible finitary linear group that is locally finite has the conjugate centralizer property with respect to any normal irreducible subgroup. Lemma 2 Let G _<__ FGL(V, K) be an irreducible, infinite dimensional and imprim- itive finitary linear group that is also locally finite. Let N be a normal irreducible subgroup of G. Then G has the conjugate centralizer property with respect to N. Proof: 12 Let U = {V.- I i E I} be a G -system of imprimitivity of V with III > 1 and let it : G —+ Sym({V}},-EI) be the permutation representation of G on U with 7r(g)(V,) = Vi". The group G/lcerrr is an infinite transitive group of finitary permutations on U (refer to Chapter 0). We define IV = N kerrr/ kerir. Since N is irreducible, IV is also an infinite transitive group of finitary permutations on U. Hence 1V is either totally imprimitive or almost primitive. In the latter case 1V contains an infinite alternating group as a section. Let F Q G be a finite subgroup of G. We denote the group errir/kerrr by F and its elements fkerir by f. The support of F = supp(F) = {V,- I Vii ¢ V} for some f E F} is finite since F is a finite set. Let 8(F) = {i E I I W}, F] ;E 0}. Using Lemma 8(ii) of [1], |8(F)| S |supp(F)| + deg(F) < oo. Define W: {Vi I i E 3(F)}. Assume first that [V is totally imprimitive. Then U is an ascending union of [V- blocks. Since W is a finite set, there exist a block B which contains W. Furthermore 3 an n E N such that if ii. = nkerrr , then Bfl n B = (D. However 8(F)n = {jEI I Vj= V,’1 for someiE%(F)} = {ieIlléeB’i}. Thus 99(F) n S‘s(F)n = (I) and by Lemmas 8(iii) and 9 of [12], [F, F"] = 1. Assume next that IV is almost primitive. Let B be a maximal IV- block and E = {B,- I i E I} be the block system of U generated by B. The permutation representation of IV on E, 7r“ : IV —> Sym(E) , is infinite, finitary and primitive. Hence 1V / kern“ is an infinite primitive finitary permutation group and so it contains the infinite alternating group (refer to Chapter 0). Define for each i E 8(F), a block B,- E S such that V,- Q B,- . Let W = {B,- I i E 8‘(F)}. We can write U as a disjoint union of W and say another set W1. Let W: {81, .82, . . . , BI} and pick any subset of W1 of t-elements say { U1, U2, . . . , Ut}. Since 1V / kervr“ contains an infinite alternating 13 group and is therefore t-transitive, there exist it E N such that B? = U,- for all i = 1,2,. . . ,t. Again 3(F) fl 8(F)n = (I) and [F, F"] = 1. C] Theorem 1 Let G g FGL(V, K) be an irreducible, infinite dimensional and im- primitive finitary linear group that is locally finite. Then G’ is the unique minimal subnormal irreducible subgroup of G. Proof: The derived group G’ of G acts irreducibly on V [Lemma 1 of [1]]. Using Lemma 2 above and (6.2) of [1], the derived group G’ is contained in every normal irreducible subgroup N of G. Furthermore if H 3 N, then H 31 G. So if M is an irreducible subnormal subgroup of G, then M 3 G. Therefore G’ Q M. D We will prove a similar version of Lemma 2 for irreducible, primitive linear groups that are locally finite. However some preliminary results are required and so we will handle the primitive groups version in Chapter 2. Recall that stable linear groups are finitary linear groups. We will prove next that certain stable linear groups have the conjugate centralizer property with respect to classical subgroups. The stable classical groups consist of the stable general linear group, the stable special linear group, the stable symplectic group, the stable orthogonal groups and the stable unitary group. We will show next that if G is the stable general linear group whose associated vector space has countably infinite dimension and whose associated field is locally finite, then G has the conjugate centralizer property with respect to its subgroups, the special general linear group, the orthogonal groups (excluding the case where the characteristic of the field is 2 and the defect of the quadratic form is 14 1) and the unitary group. If the characteristic of the field is 2 and the defect of the quadratic form is 1, then we will show that the orthogonal group acts reducibly on the associated vector space. Theorem 2 Let V be a vector space of countable dimension over a field K where K is locally finite. Then the stable linear groups Stab(GL(V,K)) have the conjugate centralizer property with respect to each of the following: 1. the stable special linear group, Stab(SL( V, K ) ); 2. the stable symplectic group, Stab(Sp( V, K ) ); 3. the stable orthogonal groups, Stab(0(V,K)), except when char(K)= 2 and the defect of quadratic form is 1, and 4. the stable unitary group, Stab( U( V, K ) ) Proof: Let {va | a Z 1} be the G-stable basis of the vector space V over the locally finite field K. The group G is a stable general linear group i.e. A 0 G = [Am 6 Gl(m,K),m 2 o 0 10, Here [00 is the countably infinite identity matrix. Let F g G be a finitely generated subgroup of G. There exist a positive integer n such that for all A E F, 0 A = where An E GL(V, K). 0 L,o We can assume that n is even. We will need the following proposition. 15 Proposition 1 Let A E Stab(GL(V, K )) be such that A = 0 In 0 Bn O B: .13: In 0 0 0 Inn 0 O In>0 Then [A,BA] = 1 and [A,BA] = andA = An 0 ’0 In 0 —In 0 0 _0 0 In 1 and so Stab( GL( V, K )) has the conjugate centralizer property with respect to any subgroup H which contains either A or A. Proof of Proposition 1_ 0 The matrix/.3 = In 0 0 In 0 En = In 0 0 0 0 0 I00 0 In 0 0 = Bn 0 0 In 0 0 I00 0 L _ .. The matrix ABA: In 0 0 BAA: 0 Bn 0 0 0 I00 Inc 00 01,0 [0 In 0 [no 100 00 [no 00 = 01,, In " 0 100 0 AnO 0 1,0 0 0 I 00 In 0 0 Bn 0 0 0 0 Bn 0 0 I 00 An 0 Bn 0 0 NCO 8 = A”. Hence BA = A—IBA ooh; = ABA. , and Thus IA, BA] = 1. The matrix A = So BA: O 0 O 16 O In 0 0 —In .1n 0 0 has inverse A"1 = In 0 0 0 I00 L 0 0 0 0 In 0 Bn 0 0 —In 0 0 0 I00 100 J 0 0 In 0 In 0 In 0 —I,, 0 0 = 0 0 = BA, 0 0 Ion L 0 I00 andasabove, IA, 3"] = [A,BA] = 1. We will show in the next lemma that either A or A is contained in almost all of the classical linear groups. Lemma 3 Let the vector space V over the locally finite field K have countably infinite dimension. Then all the classical linear groups , Stab( GL( V, K j ), Stab(SL( V, K) ), Stab(Sp(V, K)), Stab(0(V, K ,f )) and Stab(U(V, K)), except for Stab(0(V, K, f)) when the characteristic of K is 2 and the quadratic form f has defect 1, contain (relative to some basis) Proof: orA= 0 In 0 —In 0 0 0 0 I00 for all n even. case (i) : 17 Let Y = Stable general linear group. Obviously both A = In 0 0 P 0 In A = —In 0 I 0 0 case (ii) : are in Stab(GL(V,K)). Let Y = Stable special linear group, that is Since _ In case (iii) : Y: n 0 An 0 Ion 0 E SL(2n, K), A: I 0 Let Y = Stable Symplectic Group. —~In0 0 [no 0100 IAn E SL(n,K),n 2 1 The skew symmetric form on V can be represented with respect to a suitable basis X = {vn | a Z 1} by the matrix B: 1 JO 0 0 JO 0 0 J and L I o I ' J 6 Stab(SL(V,K)). where J = Then the stable symplectic group Y = {A E GL(V, K) | AtBA = —10 A and A(va) = vn for all but a finite number of vn E x} . 0 In 0 —In 0 0 I 0 0 Ion The transpose of A = A‘ O —In Hence A‘BA = In 0 0 0 O —Jn 0 0 = Jn 0 0 0 0 Jon 0 J 0 0 0 J 0 where Jn = O O _ 0 O 0 J O 0 O J 0 Thus A‘BA = 0 0 J J —In0 O 18 Note that A‘ is the transpose of matrix A . E Y where n is even. _ J00... _ 0J0 OOJ E Sp(n, K) and Joo = — We will show that A JO 0 0 JO 0 0 J Also since the matrix A has the property that A(vn) 75 vn only for aE {1,2,3,...,2n},AEY. 19 case (iv) : Y = the stable orthogonal group. The proof of this case is split into two subcases; in the first we assume char(K) 79 2 and in the second we assume char(K) = 2. (a) char(K) 96 2 : Let B be a non-degenerate symmetric bilinear form on V over K. We will prove the following: (i) : There exist a basis {u1,u2,u3, . . .} and field elements {b1,b2,b3, . . .} such that B(Ui,Uj) = bidij, b,- 75 0 for all i = 1,2,3, . ... (ii) : If K is a finite field, then B has the matrix of the form diag{1,1,1,. . ..} (iii) : If K is a locally finite field, the conclusions of (ii) still hold. Proof of (i) : There exist v E V such that B(v,v) ;£ 0. Otherwise for all u E V, 0 = B(v,v) = B(u + v,u + v) — B(u, u) — 2B(u,v). Thus B(u, v): 0 for all u,v E V and B is degenerate. Let u1 = v. Let {u1,u2, . . .} Q V be a linearly independent set with B(unuj) = bid”, b,- 75 0 V i,j E {1,2,...,k}. Define VI, = < u1,u2,...,u,c >. We will show that V = Vk EB Vki where Vki is the subspace {v E V | B(v,u,-) = 0 ‘v’ i = 1,2,3, . . . , k}. Let x E V. Choose y = x — Z,“ bT'IB(x, u,-)u,-. We now have i=1 1 B(uj,y) = B(x,uj) — bJT’B(x,uJ-)B(uj,uj) = B(x,uj) — B($,Uj) = O for allj=1,2,3,...,k. 20 Thus y E V;- and x = 21;, ble(x,u,-)u,- + y E V], + Vki. Since B is non-degenerate, B restricted to the space Vki is non—degenerate. So there exist a vector uk+1 E Vki such that B(uk+1,uk+1) 7’: 0. Let bj = B(un+1,uk+1). Since “n+1 E Vk'l’, B(unuj) = bidij for all Z,j 6 {1,2,3, . . . , k + 1}. Proof of (ii) :' Choose {u1, u2} to be linearly independent vectors of V such that B(ul, U2) = bitin- and b,- aé 0 , i, j E {1,2}. The symmetric form B restricted to V2 =< ul, u2 > is thus non-degenerate. From [5, p. 360], any non-degenerate symmetric bilinear form B on a vector space V of dimension 2 2 over a field of characteristic 31$ 2 is universal i.e. B(v, v): b has a solution for every 0 3:9 b E K. Choose ill E V2 to be such that B(u1,u1) = 1. Suppose there exist linearly independent vectors {211, 212, . . . , ilk} such that B(finiij) = 6,-3- V i,j E {1,2,...,k}. As in the proof of (i), V = Vk EB V}:- where Vk =< 211,212, . . .,i2k >. The symmetric bilinear form, B, restricted to V1,} is non- degenerate so by (i), there exist vectors of Vki, u’f and u’;, such that B(uf,u$‘) = bitin- V i, j E {1,2} and b,- 96 0. We can once again choose ilk“ E< u’f, u’2‘ > such that B(uk+1,uk+1) = 1. Thus from this construction, we obtain {711,112, . . .} such that B(v,-tn = 6,,- v i,j 2 1. Proof of (iii) : The field K is locally finite and so can be written as K = UneAKn where the Kn’s are finite fields. Let V]? be the vector space over the field K spanned by {u1, U2} and for any a E A, let Vfia be the vector space over the field Kn spanned by the same 21 vectors. From (i), we can assume that B(Ui,Uj) = biaij, b,- ¢ 0 and i,j E {1,2}. We have V1220 Q V} as subsets. We will show that B is universal on VK. Since B (u,-, Uj) = bidij, B restricted to Vfia is non-degenerate and is therefore universal. Let 0 aé f E K. There exist fl E A such that f E K ,3. Choose v E vgfl such that B(v, v) = f. But v E Vfifl Q VK and B is therefore universal on VK. Then using similar methods as in (ii), we can construct a basis {v1, v2, . . .} of V over K such that B(v,-,vj) = 6,]- V i,j 2 1. D Let A E GL(V, K) be such that A‘BA = B. By (iii), we can choose a basis x = {u1, u2, . . .} of V such that B = I with rspect to this basis. Hence A‘A = I where I is the identity matrix. Therefore Y = {A E GL(V) IA‘A = I and A(un) = un for all but a finite number of 0’s 2 1}. - I- — OInO OInO Obviously bothA= 1n 0 0 and A: _In 0 0 EY. O 0 InO 0 0 Ion (b)char(K) = 2 : For any x, y E V let B(x,y) be a symplectic scalar product on V, not necessarily non-singular. A quadratic form f on V is a function with values in K satisfying : f()\x + py) = /\2f(x) + u2f(y) + /\,uB(x, y), for all x,y E V and «Mn E K. Assume f is non-degenerate. Since K is locally finite, K is perfect ie. K = K 2 so the defect of f is 0 or 1. Let the defect of f be 0 . The matrix of B with respect to some basis 22 [l] = {61,82,63,---} is of the form 0 1 and therefore Y S Stable(Sp(V, E, K)). We will show that there exist a basis T = {61,62,63, . . .} that preserves B and such that f(e,-) = f(e._,-) = O V i _>_ 1. If for any i, f (e,-) = O, we will first show that we can also assume f (e_,-) = 0. Suppose f (8;) = 0 and f(e_,-) 75 0. Let e; = c,e,- and e’_ . ‘l = c,-e,~ +c,-'1e_,- where c, E K is such that of = (f(e_,-)). We have f(ei) = C?f(ei) = 0 and f(€—t) = Ciflez') + (6:1)2f(e—i)+ B(et, 6—2') = 0 + 1 + 1 = 0. Also B(e;,e’_,-) = B(ciei, cie, + cfle_,-) = clot-13¢“ 64) I I B(e 3') = Off-3091,60 “U! and B(e'_,-, e'_,-) B(Ciei + 0:16—23 Ciei + 01—164) = 1+1 23 Consider el and e_1. If f (el) = O, we can also assume that e_1 is also 0. Then let él = el and é_.1 = e_1. Suppose f (e1) 76 O. Wlog we can also assume f (e-1) 7’: 0. Let V1 =< e1,e_1 > and V1i = {x E V I B(e,,x) = 0 for i = :tl}. We will show that there exist :1: E V1i such that f (x) 79 0 . Suppose f (x) = O for all x E Vli. Since e2,e-2 and hence e2 + e_2 E Vli, then f(e2), f(e_2) and f(e2 + 6-2) are all 0’s. On the other hand, f(ez + 6—2) = f(62) + f(6-2) + B(eza 6—2) =1. Let x E V1L such that f(x) # 0. Let c and c1 E K be such that c2 = (f(x))‘1 and cf = (f(e1))‘1. Let e1 2 6161+ cx. We have f(eil = Cif(€1)+ 02f(33) + 6013(81, 33) = 1 + 1 + 0 =0. Let e'__1 = cf’e_1. We have B(e’l, e'_1) = B(c1e1 + cx, cl‘le_1) = 1, f(e'l) = 0 and f (e'_1) = (eff1 f (6-1) 9Q 0. Now use the technique above to produce él, é_1 such that f(e'l) = f(é_1)= 0 i.e. let (2'1 = de’l and é_1 = de’l + d‘le’_1 where cl2 = (f(eI 1)). Suppose we have {61,é_1,é2,'é_2, . . .,ék,é’_k}, linearly independent vectors of V fl .. ~ 0 if i :75 —j, such that f(e,) = 0, B(enej) = for all i,j E {:l:1,:t2,...,:l:k}. 1 iii: -—j Let V], =< éi1,éi2,...,é’ik > and Vk-L = {x E V I B(x,e',-) = 0 for all i,j E {i1,i2, . . . , ik}}. Restricting f to V?- , f is non-degenerate. As before there exist 61 and 6-1 E Vki such that f(é1)= f(e_1) = 0 and for all i,j E {1, —1}, 24 1 if 2' = —j, ~ ~ B(éi, éj) = . NOW, let 8k+1 = G1 and 6_(k+1) = é_1. 0 if i 75 —j C] Once we have bases {éih éfl, . . .} that preserves B and f (éii) = O for all i 2 1, the form f can then be described by f(x) = x1x_1 + x2x_2 + + xlx_l if x = ET rut-é}, x.- e K,l 2 1. Thus i=—l Y = {T E GL(V, K)|f(Tx) = f(x) Vx E V and T(é'i) = e, for all but a finite i's} = {T E Stable(Sp(V,T, K)) I f(Té,) = 0 Vi = :l:1, 21:2, . . .} :4 CO where n is even. 5ND Consider A = O O O N 8 Notice that f(Aéi) = f(én+i) = 0, f(Aé..+.-) = f(é.) = 0 v z“ = i1, . . Mi; f(Aéi) = f(éi) = 0 V III > 2n, and B(Aéi,Aéj) = B(éi,éj) V Z,j E Z. Thus A E Y. Also since char(K) = 2, A = A. case (iv) : Y = Stable Unitary Group. Let a : /\ H A be an automorphism of K of order 2 and H(x, y) be a non-singular hermitian product on V x V. We will show that there exist a basis x = {61, e2, . . .} SUCI’I that H(enej) = 60', i,j =3 1,2,3, . . . 25 There exist an x E V and y E V such that H (x, x) 75 0. Otherwise if x, y E V, 0 = H(x+y,x+y) = H(x,$)+H(y,y)+H($,y)+H(y.$) = H(x,y) + H(x,y)- Let oz = H(x, y). Since 0 = a + or, a = —c't. Eirthermore, 0(aa) = o(a)o(6z) =00 = ac‘t Thus ac‘t E K0, the fixed field of involution of K. We obtain eaz E K0, and since K is locally finite and therefore perfect, a E K0. Thus H (x,y) E K0 for all x,y E V. However if fl E K \Ko, H (fix, y) = flH (x, y) E K \Ko which is a contradiction. The field K is perfect so wlog we can assume H(x, x): 1 for some x E V. Let el = x. Assume next there exist linearly independent vectors in V, {e1, e2, . . . ,ek} such that H(enej) = (in, i,j E {1,2,...,k}. Let V], =< e1,e2,...,e,c >. As before V = Vk EBVl where V,“i = {v E V I H(v,e,-) = O V i = 1,2,...,k}. The hermitian product H restricted to Vki is again non-singular and one obtains as above ek+1 E Vki such that H(ek+1,ek+1) = 1. Since ek+1 E Vki, H(ek+1,e,-) = 0 Vi = 1,2,. . .,k. Thus one obtains the basis as described above. The stable Unitary group G can now be described as the set {T E GL(V, K) I Te, = e,- for all but a finite number of eIs and H(TenTej) = - 0 In 0 . H(enej) = 5U: i,j 2 1}. Consider A = In 0 0 . Since for any i and j, 00L,o d 26 Ac,- = e,-, and Aej = e,-, for some i1 and jl, H(AenAej) = H(e,,,ej,) — 611,11 = 5w = H(Ci, ej). Thus A E Y. It is also easy to check that A E Y. El Thus by Proposition 2, Stab(GL(V, K)) has the conjugate centralizer property with respect to each subgroup, Stab(SL(V,K)), Stab(Sp(V,K), Stab (O(V, K, f)) (except for when char(K) = 2 and defect of f = 1) and Stab(U(V, K)). This completes the proof of our Theorem 2. El On the other hand the stable orthogonal group, O(V, K, f), with char(K) = 2 and defect f: 1 is shown in Lemma 4 below to act reducibly on the vector space V. With this result, if M is a simple component of a group G, where G is a countable, primitive, locally finite finitary linear group of contable dimension, then M cannot be a group of this type (see chapter 2). Lemma 4 Let V be a vector space of countable dimension over a locally finite field of characteristic 2. Let G S FGL(V, K) be a countable group, G S Stab(GL(V, K )), and H g G be the stable orthogonal group, H=0( V, K, f), where the quadratic form, f, has defect 1. Then H acts reducibly on V. Proof: 27 We will first show that there exist basis A = {e1, e2, . . .} such that f (el) = 1 and f (e,-) = 0 for all i Z 2. By a suitable choice of basis, say x: {(2}, fig, . . .}, the matrix of B can be represented by . The orthogonal group Stable O(V,X, K, f): {T E GL(V)I f(Tx) = f(x) Vx E V and T(éi) = 6,- for all but a finite éi’s }. Since F is non-denegerate, f (él) 75 1. Let el = a'él where a2 = (f(é1))". Since B(é1,x) = 0 Vx E V, then B(e1,x) = O V x E V. In addition, f(e1)= a2f(é1)= 1. Use the proof from the case of the stable orthogonal group whose associated field has characteristic 2 and the the quadratic form is of defect O to produce basis elements {e2, e3, . . .} of the subspace V1 =< €52, é3, . . . > that preserves the scalar product B IV1 andf(e,-)=0Vi32. Let T E O(V, A, K, f). We will next show that T(el) = e1. Suppose T(el) = 2;, ciei. We have We) = f(Zee.) = f(eI) 28 For the next step in the argument we need to observe that B(x, y): B(Tx,Ty) for all x, y E V. This is because, f(27 + y) = f(x) + f(y) + B(x,y) f(T(x+y)) = f(Tx + Ty) = f (T17) + f (Ty) + B (Tm. Ty) = f(x) + f(y) + B(Tx,Ty), so B(x, y) = B(Tx, Ty) V x,y E V. Since B(x,e1) = O Vx E V, f(Ze-emel) = “Zea-Hue) _ = f(eI)+f(el) = 0. Also B(iqei+e1,x) = B(T(e1) +el,x) 1:] = B(T(el, x) + B(el, x) = B(Tel, x) = B(Tel, TT'1x) = B(ehT‘lx) since B(Tx,Ty) = B(x,y) = 0 Since f is non-degenerate, this implies 22;, c,-e,- + el = O and c2 2 c3 = . . . = On = 0, and CI = 1. 29 Thus K e1 is an H-subspace and so H acts reducibly on V. CHAPTER 2 Simplicity of the Derived Groups of Primitive Groups It has been observed that many results in finitary permutation groups have analogues in finitary linear groups. Here we show two such correspondences. From [8, p.10] we have the fact that every infinite transitive group of finitary permutation has the conjugate centralizer property with respect to every transitive normal subgroup of G. In this thesis, Lemma 2 of Chapter 1 and Lemma 5 below yield the following result: If G S FGL(V, K) is locally finite and dimension of V is infinite, then G has the conjugate centralizer property with respect to every irreducible normal subgroup of G. Secondly, from 4.1.3 [10] we have that if Q is infinite and G g FSym(Q, No) is primitive, then every normal subgroup of G must be transitive. It follows that G’ must be simple. Observe that if G S FGL(V, K) is primitive and dim(V) is infinite then by 4.1 of [1], every normal subgroup of G must be irreducible. Furthermore if in addition G is locally finite, Theorem 3 below produces the result that G’ is simple. Definition 7 Let G be a locally finite group. A subgroup M of G is a component ofGif (a) M<1 <1G (b) M’= M (c) M/Z(M) is simple, where Z(M) is the center of M. 30 31 In the following lemma, we will prove that if N is a simple subgroup of a locally finite and primitive finitary linear group and N is a stable group with respect to some basis A of V, then G is also stable with respect to basis A. The proof uses ideas from a recent paper by Leinen and Puglisi [7]. First we will prove the following: Proposition 2 Let G S FGL(V, K) be locally finite and primitive, N a simple sub- group of G and N S Stab(V, A, K) for some basis A of V. If V0 is a finite dimensional subspace of V, then 3 a finite subset A0 of A and a finite subgroup No of N such that the following holds: 1. V0 Q K A0 2. [A\A0,No]= 0, 3. for every v E A0, 3 g E No such that g(v) aé v and 4. N0 acts faithfully on K A0. Proof: The subspace V0 is finite dimensional so 3 a finite set A Q A such that V0 Q K A. Let A = {V,~ | i = 1, 2, . . . , k}. Since N is simple and G is primitive, N acts irreducibly on V. Thus for each i E {1,2,...,k}, 3 g, E N such that g(v,) sé v,. Let No =< g,- | i: 1,2,...,k >. Since G is locally finite, No is a finite subgroup of N. Let A0 = {v E A I g(v) at v for some g E No}. Obviously V0 Q KA Q KAo, and since No 5 N g Stab(V, A, K), A0 is finite. Furthermore, by the definition of A0, [A\A0]= 0, for each v E A0, g(v) # v for some 9 E No and No acts faithfully on KAo. [3 Lemma 5 Let G _<_ FGL(V, K) be locally finite and primitive, N a simple subgroup of G such that N g Stab(V,A, K) for some basis A of V. Then G S Stab(V, A, K). 32 Proof: Let g E G S FGL(V, K) By Proposition 1, [V, g] Q K A0 for some finite subset A0 Q A. In addition 3 a finite subgroup No Q N such that [A\Ao, N0] = 0, No acts faithfully on K A0 and for each v E A0, 3 g E No such that g(v) 51$ v. Suppose 3 g E G such that g E Stab(V, A, K) Then 3 an infinite set {v,- | i E I} Q A\Ao such that for each i E I, g(v,) 7.5 v,. We will show that for each i E I , 3 h,- e N, such that g-lh,g,-(e,-) at v,-. Since IV, g] Q K A0, we can write g(v,) = v, + w,- for some w,- E K A0. Choose h,- E No such that h,(w,-) aé w,. Therefore 9-1hi9(vi) = g‘lh,(v,-+w,-) = 9—1hi(vi) + 9-1hi(wi) g‘1(v,-) + g‘lh,(w,-) since v,- E A\Ao 9—1(vi+ hin‘l) 76 g‘1(v,- + w,) since h,(w,-) 75 w,- 121. Thus for each i E I, g‘lh,g(v,-) aé v,- for some h,- E No. But No S N is a finite subgroup and so g’lNog is also finite. Since N S G, g’lNog S N S Stab(V, A, K) and so g‘lNog centralizers all but a finite subset of A. However I is infinite and this leads to a contradiction. [3 Lemma 6 Let G S FGL(V, K) be countable, locally finite and primitive. Further- more let the dimension of V be countably infinite and K be a locally finite field. Then G has the conjugate centralizer property with respect to a simple normal subgroup of G. 33 Proof: The group G S FGL(V, K) is locally finite and primitive and the dimension of V is infinite. Thus from 10.12 of [10], G has a unique component M and CG(M) = 1. Therefore M is simple and M S G. J. Hall’s theorem [4] provides a complete list of locally finite simple subgroups of FGL(V, K) under certain restrictions. This list ‘is crucial in describing the simple subgroup M. J .Hall’s theorem states the following; Let G be a locally finite, infinite, non- “finite dimensional”, simple subgroup of FGL(V,K) where dim(V) is infinite. Then G is either alternating , the derived subgroup of FSp(V,K,a), FO(V, K, q) or FU(V, K, h) or a group of type T(W, V), W Q V" where annV(W) = 0. (See the introduction for a more thorough description of these groups.) Here a,h,q are respectively non-degenerate alternating, Hermitian and quadratic forms on V. If V has countably infinite dimension , the last group mentioned above in J. Hall’s theorem is just the group FSL(V, K). The group G S FGL(V, K) is countable and irreducible so by Lemma 1 from Chapter 1, G has a G-stable basis and thus we can regard G as a stable linear group. Furthermore if M S Stab(V, A, K) for some basis A of V, then the previous lemma showed that G S Stab(V, A, K). The derived group of GL(V, K) is SL(V, K) and the derived subgroups of FSp(V, K, a), FU(V, K, h) or FO(V, K, q) are nothing more then the intersection of SL(V, K) with each subgroup FSp(V, K, a), FU(V, K, h) or FO(V, K,q). Therefore the simple group M is one of the following; o the alternating group 0 stable SL(V, K) o SL(V, K) (1 Stable(S’p(V, K, a)) 34 o SL(V, K) (I Stable(U(V, K, h)) o SL(V, K) (1 Stable(O(V, K, q)) We will now show that G has the conjugate centralizer property with respect to M. case (i) : M = Stable SL(V,K) This follows from Theorem 2 of Chapter 1. case (ii) : M = SL(V, K) flstable(Sp(V, K, a)),SL(V, K) n stable(U(V, K, h))or SL(V, K) n stable((O, K, q)). In the case of stable((O, K, q) we assume char(K) aé 2 0 In 0 Refering back to Lemma 3 in Chapter 1, observe that A = _In 0 0 E M. 0 0 I00 Thus by Proposition 1 of Chapter 1, G has the conjugate centralizer property with respect to M . case (iii a) : M = SL(V, K) (I stable(O(V, K, q)), char(K): 2 and defect of q: 0 . n r - 0 In 0 0 In 0 Since char(K) = 2, A = 1n 0 0 _In 0 0 = A E M. Thus 0 0 Ion L 0 0 Ion again by Proposition 1, G has the conjugate centralizer property with respect to M. C] 35 case (iii b) : M = SL(V, K) n stable(O(V, K, q)), char(K): 2 and defect of q= 1 The subgroup M is the unique component of G and thus acts irreducibly on the vector space V. But Lemma 4 from the previous chapter showed that stab(O(V, K, q)) acts reducibly on V when char(K)=2 and defect q: 1. Therefore M cannot be a group of this type. case (iv) : M: the infinite alternating group. The vector space V has a G-stable basis say T = {u1,u2, . . .}. The subgroup M S G is irreducible and so the unique finitary representation of M is the natural module [12, Prop 3]. Thus with respect to M, V has the basis of the form u,- — u? where g E M. Call this basis x = {v,- | i 2 1}. We can then describe M = An 0 0 0 l 0 . . . I where An 23 a permutation matrix on {v1, v2, . . . , vn}, n 2 1 0 0 1 Let F S G be a finite subgroup. Then F moves at most {vn1 , v02, . . . , van} for some k 2 1. Let {v5,,v5,, . . . ,vgk} be disjoint from support F. The group M is k-transitive and so there exist S E M such that Svn, = vg, Vi = 1, 2,. . . ,k. Let A, B E F and let Ava, = 0a,, for i = 1,2,. . .,k. Therefore, BS(vn,) = S_’BS(vn,) : S_IB(’U3,.) = S "’(vg,) since up, is not in support(F) = ’Ua,~ 36 Therefore, ABS(vn,) = 0a,, and BSA(vn,) = BS(vni,) Thus [F, F5] = 1. Cl Observe that case (iii) above does not use the assumption that the field K is locally finite. We will show next that Lemma 5 is also true for all G S FGL(V, K) where G is locally finite and primitive, and V has infinite dimension. Proposition 3 Let G S F GL(V, K ) be locally finite and M S G be a simple ir- reducible subgroup of G. Then if F S G is a finite subgroup, 3 a simple countable subgroup of M, R, such that 1. RF = Rand 2. < R,F > acts irreducibly on a countable subspace V3,]: of V over a countable subfield KEF of K. Proof: We have M S G simple and irreducible. Thus utilizing Therem 4.1 of [9] and Lemma 12 of [12], there are countably directed systems go and iii of M such that (a) If C E p, C is a countable simple group; 37 (b) If D E R, D is countable and there is a countable subfield K D of K and a countable dimensional K D-subspace VD of V such that VD is a faithful and irreducible D-module. Further, if D, E E 5R with D Q E, then K D Q K E and VD Q VE. Let F be a finite subgroup of G, D E R and C E go. Then there is a Cl E so and a D1 E it, such that CF Q 01 Q D1. Inductively define 0,, E go and Dn E 31 by 011: g Cn+l _C_ Dn+1~ Then, if R = UDn = UCn, we see that C Q R, R is countable, RF is simple, R acts irreducibly on a countable dimensional subspace of V, and RF = R. If Y is a countable dimensional subspace on which R acts faithfully and irreducibly, then VR,F = < YF > is still of countable dimension; further < R, F > acts irreducibly on V121: and this completes the proof of the proposition. [:1 Lemma 7 Let G S FGL(V, K) be locally finite and primitive. Also let the dimension of V be infinite. Then G has the conjugate centralizer property with respect to a simple subgroup of G. Proof: Let M be the unique simple component of G. We show that G has the conjugate centralizer property with respect to M. Let F S G be a finite group. From Proposition 2 above, 3 a countable subgroup R of M such that RF = R, R is simple and < R, F > acts irreducibly on some countable subspace, VR,F of V over a countable subfield, K R, p of K. Suppose < F, R > is primitive on VER. We will first show that there exist an infinite simple subgroup N such that < F, R > has the conjugate centralizer property 38 with respect to N. If char(K) is 0, then let M R, p be the unique simple component of < F, R >. Another theorem by J. Hall [3, Theorem 1] states: Suppose G is a locally finite simple subgroup of FGL(V, K) where char(K): 0 and dimension V is infinite. Then G is the infinite alternating group. Thus M mp is an infinite alternating group and so case (iii) of Lemma 5 implies that < F, R > has the conjugate centralizer property with respect to M 3,1? ( Recall that case (iii) in Lemma 5 does not use the assumption that K is locally finite). Let N = M R,F. Suppose char(K) 9f 0. Then a theorem of Leinen [6] shows that we may assume that K is locally finite. Since the dimension of V3,]: is countably infinite, Lemma 5 implies that < F, R > has the conjugate centralizer property with respect to a simple infinite subgroup, say H. Let N = H. We will show next that N = R. Since R is simple, R n N = 1 or R n N = R. If Rfl N = 1, then NR/R ’5 N which is infinite. However, NR/R Q< F,R > /R E’ F/ F (I R which is finite. Thus R n N = R and since N is simple, R = N. Hence [F,Fg] = 1 for some g E R. But R Q M and so G has the conjugate centralizer property with respect to M. Suppose < F, R > is imprimitive on VER. Proposition 2 implies that R S < F,R > acts irreducibly on Vpfi and by Lemma 2 of Chapter 1, < F,R > has the conjugate centralizer property with respect to R. Hence G has the conjugate centralizer property with respect to M. U Theorem 3 Let G S FGL(V, K) be locally finite and primitive. Then G’ is simple Proof: 39 Again as in Lemma 6 above, let M be the unique simple component of G. Since G has the conjugate centralizer property with respect to M, we have [12, 6.2] G’ Q M. However M is simple, so G’ = M. CHAPTER 3 Composition Factors of Residually Finite Groups Theorem 4 below is the main result of this thesis. It is a generalization of a theorem by Bruno and Phillips [1, Theorem A]. The theorem by Bruno-Phillips states the following: Let G S FGL(V, K) with unip(G)= 1 and suppose that G satisfies either one of the following conditions. (i) G is both residually solvable and locally solvable (ii) G is residually finite, locally finite, and either char(K) = O , or char(K) = p > 0 and G is a p’-group. Then (a) every G-composition factor of V is finite dimensional; (b) G is a subdirect product of finite dimensional groups; (c) If the hypothesis (i) holds, G is a subdirect product of solvable groups. Our Theorem 4 generalizes the assumption of (ii). In it we only assume that G is residually finite and locally finite, with no restrictions being made on the field K. The conclusions (a) and (b) are shown to still hold. 40 41 Definition 8 An element 9 E FGL(V, K) is unipotent if (g — 1)" = 0 for some n. A subgroup H S FGL(V, K) is unipotent if each of its elements is unipotent. The subgroup unip(H) is the largest normal unipotent subgroup of H. Definition 9 A G-composition system of subspaces of V is a collection 95 = {V} | i E I} such that: 1. (l) and V E ‘39; 2. 8 is closed under unions and intersection; 3. 8 is a chain i.e. ifi,j E I, then either V, Q V,- or V,- Q V,; 4. Ifthe pair (V,,V,-) is ajump in 8 (i.e. V; Q V,, V,- 76 V,- andV, Q Vk Q V,- => k E {i, j}), then V}/V,~ is an irreducible G—module. For a G-composition system S, the factors of V are elements of the collection U = {V,~/V, I (W,V,) is a jump E 8‘} (see [12, 7.1 and Lemma 13]). Theorem 4 Let G S FGL(V, K) be such that G is both locally finite and residually finite. Then 1. every G-composition factor of V is finite dimensional and 2. G/unip( G ) is a subdirect product of finite dimensional groups. Proof of (1): The group G is residually finite. A recent paper by O. Puglisi [13, Homomorphic Images of Finitary Linear Groups, Arch. Math. 60 (1993), 497-504] showed that G / unip(G) is also residually finite. Thus wlog assume unip(G)= 1. Let L = {V,- I i E I } be a G-composition system of V. Let 11),- be the representation of G on V,- and S=fl{ker 7r,-EJ(G/kergo,-) by 17(g) = (gkCTCPj)j€J. From [1, 2.2], cp,-(g) = 1 for all but a finite number ofj E J. Also from definition of S, ker(n) is S. Thus G / S is a (restricted) subdirect product of finite dimensional groups. C] Since G is locally finite, G/S is also locally finite. From Lemma 7, G/ S is a subdirect product of finite dimensional groups. Using Proposition 1 of [1], we have 1. every transitive finitary permutation representation of G / S has finite degree. 2. every irreducible finitary representation of G is finite dimensional. The two conclusions above are exactly the hypotheses of Proposition 2 of [1] and so we conclude that S acts irreducibly on all infinite dimensional factors of L. Let J = {i E I I V,- is infinite dimensional}. Let X = $3.6le and (p, be the representation of G on V}. The group S acts faithfully and finitarily on the K-space x by sat-he; = (means for s e S and v.- e V.- For each 1 E J, let W; = €B{V,- I V,- S Vi,j E J}. Define $3 = {WI I l E J}. Observe now that every jump (WI, Wk) of L3 is such that Wk/W, 2 VI, and since S acts irreducibly on V1,, S must also act irreducibly on the jump Wk/Wl. Therefore 85 forms an S-composition of subspaces of X and L3 = {V,- I i E J} is an S-composition series of X. Define S,- = 9p,-(S) for each j E J, and note that S,- S FGL(V,,K). Let the homomorphism n : S —+ HjEJ S,- be defined by n(s) = (go,(s))jej. Since unip(G)=1 and S = fl{ker Dr{A, | i E I} x Dr{B, I j E J} be the canonical monomorphism and a,- : G —> A,, i E I and fl,- : G —-> B,, j E J be the projection maps of G onto A,- and B,- respectively. Fix It E I . Choose x E G such that x E kerak. Then 3 a minimal set 11 = {k,i1,i2,...,i,,} Q I and J1 = {j1,j2,...,j,,} Q J such that x0 Q kera, for'all i E I\Il, x0 E kerfl, for allj E J\J1 and Er : x0 —> A], x A,, x A,, x . - - x An1 x B,1 x . - - x BM is the induced monomorphism map of (1. Define Mk 2 (fl,ell\{k}kera,) fl ({1,-6J2kerfij). By minimality of 11 and J1, Mk 75 1. Since A], is simple , MkS G and an is onto, ak(Mk) = 1 or an(Mk) = A], . Suppose an(Mn) = 1. Then by definition of Mn, M], Q ker(a). Thus 1 aé Mk Q ker(a) = 1 which is a contradiction. Hence 01,,(Mk) = A], . This implies for every an E An, 39 E Mk such that fl,(g) = 1 for allj E J and a,(g) = 1 V i E I\{k} and ak(g) = an. Thus Ak Q x0 Q G Vic E I and Dr{Aka E I} Q G. Next, we will show that B = SDP{B,- I j E J} Q G. Let g E G. Then 44 (ai(g))iEI X (5j(9))j61 E G- Since Dr{At I ’i E I} Q G, a,(g"),-e; x 1 E G and so 1 x (fi,(g)),~€J Q G. Therefore B Q G. C! With the proposition above, S’ = M’ x N’. Furthermore, we conclude from the proposition that N’ is in fact a direct product of simple groups. We will next show that N=1. Since G is residually finite, then S Q G is residually finite, and so is S’. Since S = M’ x N’, then N’ is also residually finite. But N’ is a direct product of simple groups, {S,- I j E J2} Therefore N’ = 1 and N is abelian. But an irreducible representation of an abelian group is finite dimensional and the Sj’s, j E J2, are infinite dimensional finitary groups. Thus N = 1. We conclude now that S = M, a subdirect product of imprimitive groups. We will Show next that S = 1 also. To do this we will need a definition below. Definition 10 Let V be a finitary KG-module where dim(V) is infinite. A finitary K G- module WV is called an imprimitive cover of V if c VQ WV, and 0 WV has a system of imprimitivity F(WV) on which G acts transitively. The K G-module V is of imprimitive type if V has imprimitive cover. Recall that S = M is a subdirect product of {S,- I j E J1} where SIs are imprimitive groups. Thus every factor of L3 is of imprimitive type. Furthermore recall that unip(S) = 1 . Using Proposition 3 of [1], S’ is the unique minimal subnormal subgroup of S which acts irreducibly on every factor of L3 . Suppose S’ = 1; then S is abelian . But irreducible representations of abelian groups are finite dimensional. Thus J1 = (II and S = 1 Suppose S’ aé 1. The subgroup S is residually finite and so is its subgroup S’. Let x E S’ . There then exist a normal subgroup H S S’ and H 79 S’ such that x E H 45 and S’ / H is finite. By minimality of S’, H acts reducibly on some factor V}, of L3. Using (4.1) of [1], 3 an irreducible H-submodule D of VI, 3 ID, H] gé 0 and the set S = {D,- I i E I} of H -homogenous components determined by D is a system of imprimitivity for 8’. Furthermore the D,’s are finite dimensional; thus III is infinite since V], is infinite dimensional. Let 7r : S’ —> Sym(3) be the permutation representation of of S’ on 8. Then rr(S’) is a transitive group of finitary permutation and H Q kerrr. So we have the induced map it : S’ /H —> Sym(8), the permutation representation of S’ /H on 95'. Since 5’ / H is finite, III must also be finite and this is a contradiction. Hence S = 1. Recall that S = 0,6;{kercp,- I V, is finite dimensional}. Since S = 1, every G-composition factor V,, i E I is finite dimensional. Proof of (2): From 2.2 of [1], G/unip(G) is a subdirect product of FGL(V,) is the representation of G on V}, i E I . By part (1), all factors V,, i E I, are finite dimensional. Thus G / unip(G) is a subdirect product of finite dimensional groups. 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