Linear groups with the maximal condition on subgroups of infinite central dimension

Publ. Mat. 50 (2006), 103–131 LINEAR GROUPS WITH THE MAXIMAL CONDITION ON SUBGROUPS OF INFINITE CENTRAL DIMENSION L. A. Kurdachenko and I. Ya. Subbotin Abstract Let Aa vector space over a field Fand let Hbe a subgroup of GL(F, A). We define centdimFHto be dimF(A/CA(H)). We say that Hhas finite central dimension if centdimFHis finite and we say that Hhas infinite central dimension otherwise. We consider soluble linear groups, in which the (ordered by inclusion) set of all subgroups having infinite central dimension satisfies the maximal condition. Introduction Let Fbe a field, Aa vector space over F. The group GL(F, A) of all automorphisms of Aand its distinct subgroups (the linear groups) are the oldest subjects of investigation in Group Theory. The investigation of the case when Ahas finite dimension over Fwas the initial natural step. In this case, every element of GL(F, A) (a non-singular linear transformation) defines a non-singular n×n-matrix over Fwhere n= dimFA. Thus, for the finite-dimensional case the theory of linear groups is exactly the theory of matrix groups. That is why the theory of finite dimensional linear groups is one of the best developed in Algebra. However, in the case when dimFAis infinite, the situation is completely different. The study of this case requires some essential additional restrictions. The circumstances here are similar to those that appeared in the early period of development of the Infinite Group Theory. One of the most fruitful approaches here is the application of finiteness conditions to the study of infinite groups. The celebrated problem of O. Yu. ˇ Smidt regarding an infinite group with all proper subgroups finite has determined in many respects the further development of the theory of groups with finiteness conditions (see, for example, [CS]). The following two valuable generalizations follow from this: the problem of S. N. Chernikov 2000 Mathematics Subject Classification. 20F22, 20H20. Key words. Infinite dimensional linear groups, the maximal condition, soluble groups. 104 L. A. Kurdachenko, I. Ya. Subbotin on groups with the minimal condition on subgroups, and the problem of R. Baer on groups with the maximal condition on subgroups. These problems have been solved under some natural restrictions related to generalized solubility. However, in the general case, these problems have no solution yet. Moreover, A. Yu. Ol’shanski˘ı [OA, Chapter 9] has constructed a series of his brilliant examples showing that, in general, the description of Schmidt’s groups is an extremely complicated problem. We shall consider a similar problem for linear groups. As you would expect, we are trying to employ the finiteness conditions to the study of infinite dimensional linear groups. More precisely, we want to investigate such classes of linear groups that can be extracted by applying finiteness conditions on some systems of infinite dimensional subgroups. At the beginning we need to clarify the concepts of infinite dimensional and finite dimensional linear groups. If dimFAis finite, then everything is clear. If dimFAis infinite, then there are several different approaches. For instance, we can consider the following one. If His a subgroup of GL(F, A), then Hreally acts on the factor-space A/CA(H). We say that Hhas finite central dimension, if dimF(A/CA(H)) is finite. In this case dimF(A/CA(H)) will be called the central dimension of the subgroup Hand will be denoted centdimF(H). At the risk of being overly pedantic, we remark that it is crucial that His a subgroup of a particular general linear group. It is easy to construct embeddings of a group Hin two general linear groups such that Hhas finite central dimension when it is viewed as a subgroup of the first group, and has infinite central dimension as a subgroup of the second group. Consequently, we may not speak of “the class of groups of finite central dimension”. In order to avoid such kind of misunderstandings connecting with different linear representations we fix from now the concrete field F, the concrete vector space Aover F, and will consider only subgroups of GL(F, A). Let Hhas finite central dimension, then A/CA(H) is finite dimensional. Put C=CH(A/CA(H)); then, clearly, Cis a normal subgroup of Hand H/C is isomorphic to some subgroup of GLn(F) where n= dimF(A/CA(H)). Each element of Cacts trivially in every factor of series h0i ≤ CA(H)≤A, so that, Cis an abelian subgroup. Moreover, if char F= 0, then Cis torsion-free; if charF=p > 0, then C is an elementary abelian p-subgroup. Hence, in general, the structure of His defined by the structure of an ordinary finite dimensional linear group H/C. A group G≤GL(F, A)is called a finitary linear group if for each element g∈Gthe factor-space A/CA(g)has finite dimension. Finitary The Linear Groups with Max-id 105 groups are a linear analogy of the F C-groups (the groups with finite conjugacy classes). The theory of finitary linear groups is developed rather intensively and became rich with many interesting results (see, for example, the survey [PR]). This is a good example of effectiveness of finiteness conditions in the study of infinite dimensional linear groups. Some other approaches that also based on the use of finiteness conditions in infinite dimensional (near to irreducible) linear groups have been realized in [KS1], [KS2]. Let G≤GL(F, A) and let Licd(G) be the set of all subgroups of G having infinite central dimension. As the first expected step we will consider such linear groups Gclose to finite dimensional, in which the set Licd(G) is “very small” in some particular sense. According to the above analogy with the groups with finiteness conditions, the following problems logically arise: •the study of linear groups in which every proper subgroup has finite central dimension (a linear analogy of the ˇ Smidt’s problem); •the study of linear groups, in which the set of all subgroups having infinite central dimension satisfies the minimal condition (a linear analogy of the Chernikov’s problem); •the study of linear groups, in which the set of all subgroups having infinite central dimension satisfies the maximal condition (a linear analogy of the Baer’s problem). The first and the second problems have been solved for the locally soluble linear groups in the paper [DEK]. We say that a group G≤GL(F, A)satisfies the condition Max-id, if the family Licd(G)ordered by inclusion satisfies the maximal condition. In the current paper we investigate the soluble linear groups Gsatisfying Max-id. The general case immediately splits into the following two cases: •the case of groups which have no finite sets of generators; and •the case of finitely generated groups. The first case is described in the following two theorems. Theorem A (Theorem 2.6).Let G≤GL(F, A)and suppose that Gis a soluble group satisfying Max-id. If Ghas infinite central dimension and G/[G, G]is not finitely generated, then Gsatisfies the following conditions: (1) Ahas a finite series of FG-submodules h0i=C0≤C1=C≤C2· · · ≤ Cn=A 106 L. A. Kurdachenko, I. Ya. Subbotin such that dimF(A/C)is finite and Cj+1/Cjis a simple FG-module for every j,0≤j≤n−1,Q=G/CG(C)is a Pr¨ufer q-group. (2) H=CG(C1)∩CG(C2/C1)∩ · · · ∩ CG(Cn/Cn−1)is a nilpotent normal subgroup of G; moreover, if char F= 0, then His torsionfree; if char F=p > 0, then His a bounded p-subgroup. (3) G/H ≤Q×S1×· · ·×Sn−1where Qis a Pr¨ufer q-group, q6= char F, S1,...,Sn−1are finite-dimensional irreducible linear groups. In particular, Ghas the normal subgroups H≤R≤Vsuch that G/V is finite, V/R is a Pr¨ufer q-group, q6= char F,centdimF(R)is finite, R/H is finitely generated and V/H is abelian. (4) Chas an FG-submodules B=Ln∈NBnwhere each Bnis a simple FG-submodules having finite F-dimension such that C(ωFG)≤B. Here ωFG denote the augmentation ideal of the group ring FG. Theorem B (Theorem 2.9).Let G≤GL(F, A)and suppose that Gis a soluble group satisfying Max-id. If Ghas infinite central dimension and Gis not finitely generated, then Ghas a normal subgroup Ssuch that G/S is abelian-by-finite and finitely generated but S/[S, S]is not finitely generated. The case of finitely generated soluble linear groups satisfying Max-id is described in the following two theorems. In their study we will heavily use the following subgroup. Let G≤GL(F, A). Put FD(G) = {x∈G/hxihas finite central dimension}. Further we will see that FD(G) is a normal subgroup of G. Let Gbe a group, Hbe a subgroup of G, and Ube an H-invariant subgroup of G. We say that a subgroup Usatisfies the condition Max-H, if the family of all H-invariant subgroups of U, ordered by inclusion, satisfies the maximal condition. Theorem C (Theorem 2.10).Let G≤GL(F, A), and suppose that Gis a finitely generated soluble group satisfying Max-id. If Ghas infinite central dimension but centdimF(FD(G)) is finite, then the following conditions hold: (1) Ghas a normal subgroup Usuch that G/U is polycyclic. (2) There is a number m∈Nsuch that A(x−1)m=h0ifor each x∈U; in particular, Uis nilpotent. (3) If char F= 0, then Uis torsion-free, if char F=p > 0, then Uis a bounded p-subgroup. The Linear Groups with Max-id 107 (4) If h1i=Z0≤Z1≤ · · · ≤ Zm=Uis the upper central series of U, then Zj+1/Zjis a noetherian Zhgi-module for each element g∈G\FD(G),0≤j≤m−1. In particular, Usatisfies Max-hgi for each element g∈G\FD(G). Theorem D (Theorem 2.12).Let G≤GL(F, A), and suppose that Gis a finitely generated soluble group satisfying Max-id. If both centdimF(G) and centdimF(FD(G)) are infinite, then Ghas a normal subgroup L satisfying the following conditions: (1) G/L is abelian-by-finite. (2) L≤FD(G), and Lhas an infinite central dimension. (3) L/[L, L]is not finitely generated. (4) Lsatisfies Max-hgifor each element g∈G\FD(G). Finally, in the last part of the article, we consider the structure of soluble linear groups satisfying Max-id for some specific kinds of fields which make the structure of such groups more transparent. The following two results illustrate this situation. Theorem E (Theorem 3.8).Let G≤GL(F, A)and suppose that Gis a soluble group satisfying Max-id. Suppose that Gis not finitely generated. Then in each of the following cases Ghas finite central dimension: (1) Fis a field satisfying the following conditions: the Sylow q-subgroup of U(F)is finite and non-identity for each prime q, and the Sylow 2-subgroup has order at least 4. (2) Fis a field of characteristic p > 0such that the periodic part of U(F)is finite. (3) Fis a finitely generated field. (4) Fis a finite field extension of the p-adic field Qp. (5) Fis a finite field extension of a field Lwhere Lis the field of rational functions over a finite extension of Q. Here U(F) denotes the multiplicative group of a field F. Theorem F (Theorem 3.9).Let G≤GL(F, A)and suppose that Gis a soluble group satisfying Max-id. Suppose that Gis finitely generated. Then in each of the following cases the finitary radical of Ghas finite central dimension: (1) Fis a field satisfying the following conditions: the Sylow q-subgroup of U(F)is finite and non-identity for each prime q, and the Sylow 2-subgroup has order at least 4. 108 L. A. Kurdachenko, I. Ya. Subbotin (2) Fis a field of characteristic p > 0such that the periodic part of U(F)is finite. (3) Fis a finitely generated field. (4) Fis a finite field extension of the p-adic field Qp. (5) Fis a finite field extension of a field Lwhere Lis the rational functions field over a finite extension of Q. 1. Preliminary results Lemma 1.1. Let G≤GL(F, A). (i) If L≤H≤Gand centdimF(H)is finite, then centdimF(L)is also finite. (ii) If Hand Lhave finite central dimension, then centdimF(hH, Li) is likewise finite. In fact, if centdimF(H) is finite, then CA(H) has a finite codimension. Thus if Lis a subgroup of H, then CA(L)≥CA(H); so that, CA(L) has finite codimension too. If Hand Lhave finite central dimension, then the both subspaces CA(H) and CA(L) have finite codimension. It follows that dimF(A/(CA(H)CA(L)) is finite too. Corollary 1.2. Let G≤GL(F, A). Then the set FD(G) = {x∈G/hxihas finite central dimension} is a normal subgroup of G. In fact, by Lemma 1.1 FD(G) is a subgroup. Let x∈FD(G), g∈G. Since CA(xg) = CA(x)g,CA(xg) also has finite codimension. Note that G=FD(G) if and only if Gis a finitary linear group. Therefore the subgroup FD(G)is called the finitary radical of a linear group G. Lemma 1.3. Let G≤GL(F, A)and suppose that Gsatisfies Max-id. (i) If His a subgroup of G, then Hsatisfies Max-id. (ii) If H1< H2<··· < Hn < ··· is a chain of subgroup, then each subgroup Hnhas finite central dimension. (iii) If Hhas infinite central dimension, then an ordered by inclusion set L[H, G]of all subgroups containing H, satisfies the maximal condition. (iv) Either every finitely generated subgroup of Ghas finite central dimension or Gis a finitely generated group. In particular, if Gis not finitely generated, it is a finitary linear group. The Linear Groups with Max-id 109 Proof: (i) is obvious. To prove (ii) we note that there is a number d such that every subgroup Hnhas finite central dimension for n≥d. By Lemma 1.1 the subgroups H1,...,Hd−1have also finite central dimensional; and (iii) is also obvious. Finally, we will prove (iv). Suppose that Ghas no finite sets of generators. Let Lbe a finitely generated subgroup of G. Then G\L6=∅. Let a1∈G\L,L1=hL, a1i. Since L1is finitely generated, G6=L1; that is, G\L16=∅. Using the similar arguments, we can construct a strictly ascending series L < L1<··· < Ln<··· of finitely generated subgroups. By Lemma 1.1 every subgroup Lnhas finite central dimensional, in particular, centdimF(L) is also finite. Corollary 1.4. Let Gbe a group satisfying Max-id,Ha subgroup of G, and Ka normal subgroup of H. If H/K does not satisfies the maximal condition, then Khas finite central dimension. Corollary 1.5. Let Gbe a group satisfying Max-id,Ha subgroup of G and Ka normal subgroup of H. If H/K =Drλ∈Λ(Hλ/K)where Hλ6=K for every λ∈Λ, and the set Λis infinite, then Khas finite central dimension. Corollary 1.6. Let Gbe a group satisfying Max-id,Ha subgroup of Gand Kan H-invariant subgroup of G. If centdimF(H)is infinite and H∩Ksatisfies the maximal condition for subgroups, then K satisfies Max-H. Proof: If Ksatisfies Max (the maximal condition for all subgroups), then all is proved. Therefore, suppose that Khas a strictly ascending series L1<···< Ln<··· of H-invariant subgroups of K. Consider the ascending chain HL1≤ · · · ≤ HLn≤ · · · . Suppose that there exists a positive integer msuch that HLm=HLn for all n≥m. Since H∩Ksatisfies Max, there exists a positive integer t such that H∩Lt=H∩Lnfor all n≥t. Let d= max{m, t}and n≥t. The inclusion Lt≤Lnimplies Ln=Lt(H∩Ln) = Lt(H∩Lt) = Lt. This fact contradicts to our assumption concerning the chain {Ln|n∈N}. This contradiction shows that the chain {HLn|n∈N}is strictly ascending. It follows that there exists a positive integer ksuch that HLt has finite central dimension. By Lemma 1.1 centdimF(H) is finite. 110 L. A. Kurdachenko, I. Ya. Subbotin Lemma 1.7. Let Gbe a group satisfying Max-id,Ha subgroup of G and Ka normal subgroup of H. If centdimF(K)is infinite, then H/K satisfies the maximal condition. In particular, if His locally (soluble-byfinite), then H/K is polycyclic-by-finite. Lemma 1.8. Let G≤GL(F, A)and suppose that Gsatisfies Max-id. Suppose also that Land Hare subgroups of Gsatisfying the following properties: •L=Drλ∈ΛLλwhere Lλis a non-identity H-invariant subgroup of Lfor every λ∈Λ; and •H∩L≤Drλ∈MLλ. If the set Γ = Λ Mis infinite, then the subgroup H has finite central dimension. Proof: Since Γ is infinite, it has an infinite ascending chain Γ1⊆Γ2⊆ · · · ⊆ Γn⊆ · · · of infinite subsets. Since H∩Drλ∈ΛLλ=h1i, we come to an ascending chain of subgroups hH, Lλ|λ∈Γ1i<hH, Lλ|λ∈Γ2i<···<hH, Lλ|λ∈Γni<··· . There is a number dsuch that the subgroup hH, Lλ|λ∈Γdihas finite central dimension. By Lemma 1.1 Halso has finite central dimension. Lemma 1.9. Let G≤GL(F, A)and suppose that Gsatisfies Max-id. Let Hand Qbe subgroups of Gsatisfying the following conditions: •Qis a normal subgroup of H; and •H/Q =B/Q ×C/Q. If the subgroups B/Q and C/Q do not satisfy the maximal condition, then the subgroup Hhas finite central dimension. Proof: Since H/B ∼ =C/Q, it does not satisfies Max. Corollary 1.4 yields that centdimF(B) is finite. By the same reasons centdimF(C) is finite. Since H=BC, Lemma 1.1 implies that Hhas finite central dimension as well. Corollary 1.10. Let G≤GL(F, A)and suppose that Gsatisfies Max-id. Let Hand Qare the subgroups of Gsatisfying the following conditions: •Qis a normal subgroup of H; and •H/Q =Drλ∈Λ(Lλ/Q)where Lλ6=Qfor every λ∈Λ. If the set Λis infinite, then centdimF(H)is finite. The Linear Groups with Max-id 111 Proof: There are two subsets Γ and ∆ of Λ with the following properties: Γ∪∆ = Λ, Γ∩∆ = ∅, Γ, ∆ are infinite. It follows that Γ (respectively ∆) has an infinite ascending chain Γ1⊆Γ2⊆ · · · ⊆ Γn··· (respectively ∆1⊆∆2⊆ · · · ⊆ ∆n⊆ · · · ) of infinite subsets. Put U/Q =Drλ∈Λ(Lλ/Q), V/Q =Drλ∈Λ(Lλ/Q). Then we obtain two ascending chains of subgroups hLλ|λ∈∆1i<hLλ|λ∈∆2i<···<hLλ|λ∈∆ni<··· and hLλ|λ∈Γ1i<hLλ|λ∈Γ2i<···<hLλ|λ∈Γni<··· . It follows that the groups U/Q and V/Q do not satisfy Max. Since H/Q= U/Q ×V/Q, Lemma 1.9 gives that Hhas finite central dimension. Lemma 1.11. Let Gbe a soluble subgroup of GL(F, A)and suppose that Gsatisfies Max-id. Then G/FD(G)is polycyclic. Proof: Let h1i=D0≤D1≤ · · · ≤ Dn=Gbe the derived series of G. If Gis not finitely generated, then Lemma 1.3 yields that G=FD(G). Suppose that Gis finitely generated. Then G/Dn−1is finitely generated. If Dj+1/Djis finitely generated for each j, 0 ≤j≤n−1, then Gis polycyclic. Therefore, we may assume that there is a number m∈Nsuch that G/Dmis polycyclic and Dm/Dm−1is not finitely generated. In particular, Dmis not finitely generated and Dm≤FD(G) by Lemma 1.3. 2. The general structure of the soluble linear groups satisfying Max-id Recall that a group Gis said to be quasicyclic or a Pr¨ufer p-group, where pis a prime, if G=han|ap 1= 1, an+ 1p=an, n ∈Ni. A group Gis said to be a Chernikov group, if Ghas a normal subgroup of finite index, which is decomposed into a direct product of finitely many Pr¨ufer groups. Lemma 2.1. Let G≤GL(F, A)and suppose that Gsatisfies Max-id. If Ghas infinite central dimension and G6= [G, G] = D, then either Gab = G/D is finitely generated or it has a finitely generated subgroup S/D such that G/S is a Pr¨ufer p-group for some prime p. Proof: Suppose that G/D is not finitely generated. Let T/D be the periodic part of G/D. Choose in Q=G/T a maximal Z-independent set of elements {uλ|λ∈Λ}. Then the subgroup U=huλ|λ∈Λi 118 L. A. Kurdachenko, I. Ya. Subbotin normal abelian p-subgroup, Qis the group of q-adic numbers, p,qare primes, and p6=q. In the paper [MPP] U. Meierfrankenfeld, R. E. Phillips and O. Puglisi considered the locally soluble finitary linear groups. In particular, they proved that locally soluble finitary linear groups is unipotent-by-abelianby-(locally finite). Since we consider more concrete cases, we are able to get more complete description of these factors. The next natural step is consideration of finitely generated case. Theorem 2.10. Let G≤GL(F, A), and suppose that Gis a finitely generated soluble group satisfying Max-id. If Ghas infinite central dimension but centdimF(FD(G)) is finite, then the following conditions hold: (1) Ghas a normal subgroup Usuch that G/U is polycyclic. (2) There is a number m∈Nsuch that A(x−1)m=h0ifor each x∈U, in particular, Uis nilpotent. (3) If char F= 0, then Uis torsion-free, if char F=p > 0, then Uis a bounded p-subgroup. (4) If h1i=Z0≤Z1≤ · · · ≤ Zm=Uis an upper central series of U, then Zj+1/Zjis a noetherian Zhgi-module for each element g∈G\FD(G),0≤j≤m−1. In particular, Usatisfies Max-hgifor each element g∈G\FD(G). Proof: Put C=CA(FD(G)). Since dimF(A/C) is finite, Ahas a finite series of FG-submodules h0i=C0≤C1=C≤C2≤ · · · ≤ Cm=A such that C2/C1,...,Cm/Cm−1are simple FG-modules having finite dimension over F. By a Maltsev’s Theorem (see, for example, [WB, Lemma 3.5]) G/CG(C2/C1), . . . , G/CG(Cn/Cn−1) are abelian-by-finite groups and hence polycyclic, because Gis finitely generated. Put U=CG(C1)∩CG(C2/C1)∩ · · · ∩ CG(Cm/Cm−1). Clearly, CG(C1)≥FD(G), so G/CG(C1) is polycyclic by Lemma 1.1. Then the embedding G/U ≤G/CG(C1)×G/CG(C2/C1)× · · · × CG(Cn/Cn−1) proves that G/U is polycyclic. Each element of Uacts trivially in every factor Cj+1/Cj, 0 ≤j≤m−1. It follows that Uis a nilpotent subgroup, moreover, if char F= 0, then His torsion-free; if char F=p > 0, then His a bounded p-subgroup (see, for example, [KW, Proposition 1.C.3] and [FL, Section 43]). The assertion (4) follows from Corollary 1.6. The example below helps us to illustrate these results. The Linear Groups with Max-id 119 Example 2.11. Let Fbe a field, which is not locally finite, and put A=Ln∈NAnwhere An∼ =Ffor each n∈N. Choose in U(F) an element gof infinite order and consider the following infinite matrix γ=kujmkj,m∈N, where ujj =gjfor all j∈Nand that ujm = 0 for j6=m. Consider now the set Σ of all matrices α=kujmkj,m∈Nsuch that ujm = 0 if (j, m)/∈ {(1,2),(j, j)|(j∈N}. If β=kvjmkj,m∈Nis the other matrix of Σ, then αβ =kwjmkj,m∈N where wjj =ujjvjj ∈N, w12 =u11v12 +u12v22, wjm = 0 if (j, m)/∈ {(1,2),(j, j)|j∈N}. It follows that αβ ∈Σ. Furthermore, if α−1=kyjmkj,m∈N, then yjj = u−1 jj ,j∈N,y12 =u−1 11 v−1 22 u12, in particular, α−1∈Σ. This means, that Σ is a subgroup of GL(F, A). Consider now the set Φ of all matrices α= kujmkj,m∈Nsuch that ujj = 1 for all j∈N,ujm = 0 if (j, m)/∈ {(1,2)}. If β=kvjmkj,m∈Nis another matrix of Φ, then αβ =kwjmkj,m∈N where wjj = 1 for all j∈N,w12 =v12 +u12. It follows that α, β ∈Φ. Moreover, Φ is isomorphic to the additive group of the field F. Let α=kujmkj,m∈N∈Σ, β =kvjmkj,m∈N∈Φ then α−1βα =kwjmkj,m∈Nwhere wjj = 1 for all j∈N,w12 = u−1 11 u22v12. It follows that Φ is a normal subgroup of Σ. Let τ= kujmkj,m∈Nsuch that ujj = 1 for all j∈N,u12 = 1, and ujm = 0 if (j, m)/∈ {(1,2)}. Put Γ = hγ, τi. Then Γ = T⋊hγiwhere Tis isomorphic to the subgroup of the additive group of Fgenerated by the elements {gn|n∈Z}. We can consider Tas a Zhγi-module. Since Zhγiis a noetherian ring, a cyclic Zhγi-module is noetherian. It is easy to see that Γ satisfies Max-id. We have considered the easiest case. However, the construction above allows the normal unipotent subgroup Thas as much as desired length of nilpotency. Theorem 2.12. Let G≤GL(F, A), and suppose that Gis a finitely generated soluble group satisfying Max-id. If centdimF(G)and centdimF(FD(G)) are infinite, then Ghas a normal subgroup Lsatisfying the following conditions: (1) G/L is abelian-by-finite. (2) L≤FD(G), and Lhas an infinite central dimension. (3) L/[L, L]is not finitely generated. (4) Lsatisfies Max-hgifor each element g∈GFD(G). 120 L. A. Kurdachenko, I. Ya. Subbotin Proof: Let h1i=D0≤D1≤ · · · ≤ Dn=G be the derived series of G. If Gis polycyclic, then FD(G) is likewise polycyclic, and Lemma 1.1 proves that dimF(A/CA(FD(G)) is finite. This contradiction shows that there is a number m∈Nsuch that G/Dmis polycyclic and Dm/Dm−1is not finitely generated. Repeating the arguments of the proof of Theorem 2.9, we can construct a normal subgroup Lsuch that G/L is abelian-by-finite and L/[L, L] is not finitely generated. In particular, Lis not finitely generated, so that L≤FD(G) by Lemma 1.3. If we suppose that dimF(A/CA(L)) is finite, then Lemma 1.1 yields that dimF(A/CA(FD(G))) is finite, because FD(G)/L is finitely generated. Finally the assertion (4) follows from Corollary 1.6. 3. The structure of the soluble linear groups with Max-id over some specific fields The structure of finite dimensional soluble linear groups is often defined by the structure of the multiplicative group of the field, over which such groups are considered. More precisely, it is defined by not only the structure of this multiplicative group, but also by the structure of multiplicative groups of finite extensions of the base fields. So it is reasonable to expect, that the same dependence takes place also for infinite dimensional linear groups satisfying Max-id. Proposition 2.4 and Theorem 2.6 logically lead us to the consideration of the following types of fields. If Gis an abelian subgroup, then as usually, we will denote by t(G) the maximal periodic subgroup of G, the periodic part of G. Lemma 3.1. Let Gbe a soluble subgroup of GL(F, A)satisfying Max-id. Suppose that the field Fsatisfies the following condition: (RE) for each finite field extension Eof F(in particular, for E=F) the subgroup t(U(E)) has only finite Sylow q-subgroup for each prime q. Then either Ghas finite central dimension or G/[G, G]is finitely generated. Proof: Put D= [G, G], and suppose that dimF(A/CA(G)) is infinite and G/D is not finitely generated. Lemma 2.1 yields that Ghas a normal subgroup L≥Dsuch that L/D is finitely generated and G/L is a Pr¨ufer q-group for some prime q. By Lemma 1.3 Gis a finitary linear group. Corollary 1.4 yields that Lhas finite central dimension; so that, The Linear Groups with Max-id 121 C=CA(L) has finite codimension. Furthermore, CG(C)≥L; in particular, either G=CG(C) or G/CG(C) is a Pr¨ufer q-group. In a first case, centdimF(G) is finite. Hence, G/CG(C) is a Pr¨ufer q-group. Lemma 5.1 of [DEK] proves that q6= char F. Since Lis a normal subgroup of G, Cis an FG-submodule of A. Put G/CG(C) = hgnCG(C)|gp 1∈CG(C), gp n+1 ∈gnCG(C), n ∈Ni. Since G/CG(C) is abelian, Yn=CA(gn) = CA(hgni) is an FG-submodule of Cfor each n∈N. By Lemma 1.3 Gis a finitary linear group; so that, dimF(C/Yn) is finite for each n∈N. We have already noted that q6= char F, therefore by Maschke’s Theorem (see, for example, [WB, Corollary 1.6]) C/Yn=M1/Yn⊕ · · ·⊕Mk/Yn, where Mj/Ynis a simple FG-submodule of finite F-dimension. By Proposition 2.4 G/CG(Mj/Yn) is isomorphic to a subgroup of U1× · · · × Unwhere Ujis isomorphic to a multiplicative group of a field Efor certain finite field extension E of the field F, 1 ≤j≤n. Since CG(Mj/Yn)≥CG(C), then either G=CG(Mj/Yn) or G/CG(Mj/Yn) is a Pr¨ufer q-group. By our condition (RE) the group U(E) does not have a Pr¨ufer q-subgroup. This implies that G=CG(Mj/Yn) for each n∈N. In turn, it follows that G=CG(C/Yn). Since it is valid for each n∈N,G=∩n∈NCG(C/Yn) = CG(C). Thus we again obtain that Ghas finite central dimension. This contradiction proves our lemma. Proposition 3.2. Let Gbe a soluble subgroup of GL(F, A)satisfying Max-id and suppose that a field Fsatisfies the condition (RE). (1) If Gis not finitely generated, then Ghas finite central dimension. (2) If Gis finitely generated, then FD(G)has finite central dimension. Proof: Let h1i=D0≤D1≤ · · · ≤ Dk=G be the derived series of G. If every factor of this series is finitely generated, then Gsatisfies Max. Therefore assume that there is a number t such that G/Dtis polycyclic, but Dt/Dt−1is not finitely generated. In particular, G=hDt, Sifor some finite subset S. By Lemma 3.1 dimF(A/CA(Dt)) is finite. If Gis not finitely generated, then Lemma 1.3 shows that Gis a finitary group and the equation G=hDt, Sicoupled with Lemma 1.1 imply that Ghas finite central dimension. If Gis finitely generated, then Dt≤FD(G). Since FD(G)/Dtis finitely generated, Lemma 1.1 proves that FD(G) has finite central dimension. R. M. Guralnick kindly provided us with the following results about the structure of the multiplicative groups of some types of fields. 122 L. A. Kurdachenko, I. Ya. Subbotin Proposition 3.3. Let Fbe a field and suppose that the Sylow q-subgroup of U(F)is finite and non-identity for each prime q, and the order of the Sylow 2-subgroup is at least 4. If Eis a finite field extension of F, then the Sylow q-subgroup of U(E)is finite for each prime q. Proof: Let Sq(respectively Rq) be a Sylow q-subgroup of U(F) (respectively U(E)). Then by Lemma 4.2 of [GW]Rq/Sqis finite. It follows that Rqis finite for each prime q. The multiplicative groups of rings considered in Proposition 3.3 have a very large (although reduced) periodic parts. In this connection, it will be interesting to consider the dual situation; that is, the case when the periodic part of the multiplicative group of a field is finite. But the example of the complex field C, which is a finite extension of the field R, shows that not every field Ffor which t(U(F)) is finite satisfies the condition (RE) (note that t(U(R)) has order 2, but t(U(C)) is divisible). Note that in this example we are dealing with the fields of characteristic zero. The following theorem shows that for fields of positive characteristic we have much better situation. Theorem 3.4. Let Fbe a field of characteristic p > 0and suppose that the periodic part of U(F)is finite. If Eis a finite field extension of F, then the periodic part of U(E)is likewise finite. Proof: Let Pbe the prime subfield of F. We may assume that Eis a normal extension of F(this just enlarges E). We may also assume that extension Eis separable (if not, we can replace Eby F(T) where T= t(U(E)); this is a separable Galois extension). Let Gbe the Galois group of this extension. Consider E∗=P(T). Then F∗=F∩E∗=P(F∩T) and by assumption, this is a finite field. We also see that F∗is the fixed field of Gacting on E∗and so E∗is a finite Galois extension of F∗and so E∗is finite, whence the periodic part of U(E) is finite. Further, if Fis a field, the multiplicative group of which is residually finite, then every Sylow q-subgroup of U(F) is finite for each prime p. In connection with Proposition 3.2 the following question is naturally raised: is a Sylow q-subgroup of the multiplicative group of an arbitrary finite field extension of Ffinite for each prime q? In particular, is the multiplicative group of an arbitrary finite field extension of Fresidually finite? R. M. Guralnick provided us with the following counterexamples for this question. The Linear Groups with Max-id 123 Proposition 3.5. There exist fields Fand Esatisfying the following conditions: (1) char F= 0. (2) U(F)is residually finite and t(U(F)) is a finite subgroup of order 2. (3) Eis a finite field extension of F. (4) U(E)has a Pr¨ufer q-subgroup for some prime q, in particular, U(E)is not residually finite. Proof: Let qbe a prime. Let Sbe a Pr¨ufer q-subgroup of U(C) and put E=Q(S). Clearly U(E) has S, in particular, U(E) is not residually finite. Let Fbe the fixed field of Eunder complex conjugation. So [E:F] = 2. We show that U(F) is residually finite, the periodic part of U(F) is finite (has order 2), but, as we have recently noted, U(E) has a Pr¨ufer q-subgroup. Since Rcontains F, the periodic part of U(F) has order 2. So U(F) = T×Vwhere |T|= 2, Vis a torsion-free subgroup (see, for example, [FL, Theorem 27.5]). Hence it suffices to prove that Vis residually finite. Since V/V nis bounded, it is decomposed into a direct product of finite cyclic groups by Pr¨ufer’s Theorem (see, for example, [FL, Theorem 17.2]). If we prove that ∩n∈NVn=h1i, it will implies that Vis residually finite. Let 1 6=v∈V. Since v∈E,vis algebraic over Q, so Q(v) is a finite extension of Q. Then U(Q(v)) is a direct product of a finite cyclic subgroup and a free abelian subgroup (see, for example, [KG, Chapter 4, Corollary 5.7]). Thus we can choose a sufficiently large prime rsuch that v /∈(Q(v))r. Suppose that v∈Vr, then there must exists an element w∈Fsuch that v=wr. Then Q(w) is a nontrivial abelian extension of Q(v) (since Eis generated over Qby roots of unity). The field Q(w) must contain at least two roots of a polynomial Xr−v and their ratio is an r-th root of 1. This implies that Q(w) is not fixed by complex conjugation and so wis not in F. This contradiction shows that v /∈Vr, as required. In this example the periodic part of the multiplicative group of a basic field is very small —the least possible. In the following proposition the multiplicative group is periodic. Proposition 3.6. There exists fields Fand E satisfying the following conditions: (1) Fand Eare locally finite fields. (2) For each prime rthe Sylow r-subgroup of U(F)is finite. 124 L. A. Kurdachenko, I. Ya. Subbotin (3) Eis a finite field extension of F. (4) There exists a prime qsuch that the Sylow q-subgroup of U(F)is identity, but the Sylow q-subgroup of U(E)is a Pr¨ufer q-subgroup, in particular, U(E)is not residually finite. (5) For each prime r6=qthe Sylow r-subgroup of U(E)is finite. Proof: Let pbe a prime and qa distinct prime with GCD(q, p −1) = 1. Let Fbe the union of the finite fields of size pmewhere me=qe,e∈N. By the choice of qthe Sylow q-subgroup of Fis identity. Let E=F[a] where a is a q-th root of 1 and let E0be the subfield of Egenerated over the prime subfield. Then E0is finite, say |E0|=pd. So pd≡1 (mod q). Then Econtains a subfield of degree se=dqeover the prime field for each e∈N. However, qedivides pse−1 for all e∈N and so the Sylow q-subgroup Eqof U(E) is infinite. This means that Eqis a Pr¨ufer q-subgroup. We note now that the Sylow r-subgroup Frof U(F) is finite for each prime r. If rdivides pme−1 with eminimal, then we pick no more powers of rfor any larger e, and so U(F) is residually finite. However, U(E) is not residually finite, because it has a Pr¨ufer q-subgroup. We prove now that that the Sylow r-subgroup Erof U(E) is finite for each prime r6=q. Choose the smallest seso that the field of size pse contains r-th roots of 1 (or 4-th roots of 1 if r= 2; if no such field exists, we are done); we are then passing to a series of extensions of degree q and we pick up no further r-torsion (see Lemma 4.2 in [GW]). The following theorem is concerned with the class of fields for which residual finiteness of the multiplicative group is inherited by every finite field extension. Theorem 3.7. Let Fbe a field with a discrete valuation vand Kbe the residue field. Suppose that the following conditions hold: (1) Khas positive characteristic. (2) U(P)is residually finite for every finite extension Pof K(in particular, for P=K). If Eis a finite field extension of F, then U(E)is residually finite. Proof: Let char K=p. There is no harm in taking Eis a normal extension of F. It is straightforward to reduce to the case that Eis a separable extension of F(and so Galois) (if vis an n-th power for all n, the same is true for vp; and so, we may take vto be separable over F —then the vis a papower for every a, and if some such element were not separable over E, they would generate larger and larger extensions). The Linear Groups with Max-id 125 Now, since Eis separable extension of F, we see that the discrete valuation has only finitely many extensions to E, which are all discrete. So if v∈U(E) is an n-th power for all n, the valuation of vmust be zero. The condition on the residue fields implies that v= 1 modulo each maximal ideal over the maximal ideal of F. It is straightforward to see that such elements (other than 1) are not papowers for a sufficiently large. Now we will consider fields with some other properties which imply additional conditions on linear groups with Max-id over such fields. Let Fbe a field. We say that Fhas a property (FAE) if for each finite field extension Eof F(in particular, for E=F) the factorgroup U(E)/t(U(E)) is free abelian. Lemma 3.8. Let Gbe a subgroup of GL(F, A)satisfying Max-id. Suppose that Ghas subgroups Vand Hwith the following properties: (1) His a normal subgroup of V. (2) His a nilpotent bounded p-subgroup for some prime p. (3) V/H is a Pr¨ufer q-group, q6=p. (4) dimF(A/CA(V)) is infinite. Then Hhas a finite V-composition series. Proof: By Lemma 1.D.4 of the book [KW]V=H⋊Qwhere Qis a Pr¨ufer q-subgroup, so that, Q=hzn|zp 1= 1, zp n+1 =zn, n ∈ Ni. If His finite, then all is proved. Suppose that His infinite. It follows that H/[H, H] is infinite (see, for example, [RD1, Corollary of Theorem 2.26]). Corollary 1.5 yields that dimF(A/CA(H)) is finite. The equation V=HQ combined with Lemma 1.1 imply that dimF(A/CA(Q)) is infinite. Let C,Dbe the V-invariant subgroup of H such that D≤C, and B=C/D is an elementary abelian p-group. If zis an arbitrary element of Q, then by Maschke’s Theorem (see, for example, [WB, Theorem 1.5]) B=Lλ∈ΛBλwhere Bλis the minimal hzi-invariant subgroup of Bfor each λ∈Λ. In particular, either [Bλ, z] = Bλ or [Bλ, z] = h1i. It follows that B=CB(z)×[B, z]. Since Qis abelian, CB(z) and [B, z] are the Q-invariant subgroups of B. If yis an element of Qsuch that hzi ≤ hyi, then CB(y)CB(z), and [B, z]≤[B, y]. Thus we have an ascending series [B, z1]≤[B, z2]≤ · · · ≤ [B, zn]≤ · · · 126 L. A. Kurdachenko, I. Ya. Subbotin of Q-invariant subgroups. By Corollary 1.6 there is a number m∈N such that [B, zn] = [B, zm] for all n≥m. The equation B=CB(zj)× [B, zj] implies that CB(zn) = CB(zm) for all n≥m. In other words, Bsatisfies the minimal condition for centralizers. By Theorem A of paper [HB1]Bis a semisimple FpQ-module. By Corollary 1.6 Bsatisfies Max-Q, hence Bis a direct sum of minimal Q-invariant subgroups. In turn, it follows that Hhas finite Q-composition series, because His nilpotent and bounded. Proposition 3.9. Let Fbe a field of positive characteristic pand Gbe a soluble subgroup of GL(F, A)satisfying Max-id. Suppose that Ghas infinite central dimension and is not finitely generated. If Fsatisfies (FAE), then (1) Ghas the normal subgroups H≤Rsuch that G/R is polycyclic, R/H is a Pr¨ufer q-group, q6= char F. (2) His a nilpotent bounded p-subgroup. (3) Hhas a finite G-composition series. Proof: Suppose first that G/[G, G] is not finitely generated. Using Theorem 2.6, we obtain that Ghas a normal nilpotent bounded p-subgroup H, such that G/H ≤Q×S1× · · · × Sn−1where Qis a Pr¨ufer q-group, q6= char F,S1,...,Sn−1are ordinary finite-dimensional irreducible linear groups. By Proposition 2.4 Sjhas a normal subgroup of finite index, which is isomorphic to some subgroup of U1× · · · × Umwhere Utis isomorphic to the multiplicative group of a field Efor certain finite field extension Eof the field F, 1 ≤t≤m. Since Fsatisfies (FAE), Ut/t(Ut) is free abelian for each t, 1 ≤t≤m. It follows that Ghas a normal subgroup R≥Hsuch that G/R is abelian-by-finite and finitely generated, R/H is a Pr¨ufer q-group, q6= char F. Since Gis not finitely generated, Lemma 1.3 proves that Gis a finitary linear group. Lemma 1.1 yields that Rhas infinite central dimension. By Lemma 3.10 Hhas finite R-composition length, therefore Hhas finite G-composition length. For the general case, it is sufficient to apply Theorem 2.9. Now we will consider some particular fields satisfying (FAE). Proposition 3.10. Let Fbe a finitely generated extension of locally finite field L. Then U(F)/t(U(F)) is free abelian. The Linear Groups with Max-id 127 Proof: Note that the property (FAE) is valid for any subfield. Thus, there is no harm in replacing Lby Kwhere Kis the algebraic closure of L (and is still locally finite) and Fby FK =P(note that the hypothesis still hold). Now Pis a finite extension of K(x1,...,xr) where the xjare algebraically independent, 1 ≤j≤r. We can apply Noether’s Lemma and see that we may choose the xjso that Pis the quotient field of the integral closure Dof K[x1, . . . , xr]. Let Vbe the set of discrete valuations corresponding to the places of D. Then we have the map 1 →U(D)→U(P)→Gwhere G∼ = Lv∈VYvand Yv∼ =Z,v∈V, and Vis the set of valuations on P(given by u→Pv(u)). Now it is well known that U(D)/U(K) is a finitely generated abelian [GW], and the image of the map is contained in the free abelian group Gand so is also free abelian. Thus, the map splits and we see that U(P)/U(K)∼ =A⊕Bwhere Ais finitely generated and Bis free abelian. Since U(K) is the periodic part of U(P), the result is proved. The following lemma gives us an example of a field of characteristic zero, satisfying (FAE). Lemma 3.11. Let Fbe a field and suppose that E=F(X|λ∈Λ) is a field extension of Fsuch that {X|λ∈Λ}are algebraically independent. Then U(E)∼ =U(F)×Bwhere Bis free abelian. Proof: We have the obvious F-valuations on E(corresponding to irreducible polynomials). This gives a map 1 →U(D)→U(E)→Awhere Ais free abelian. Thus, U(E)∼ =U(F)×Bwhere Bis the image of the map and so free. Proposition 3.12. Let Lbe a rational functions field over a finite extension of Q, and let Fbe a finite extension of L. Then U(L)/T is free abelian, where Tis the periodic part of U(L). Moreover, Tis finite. Proof: We have F=L(a) for some element a. The minimal polynomial of ainvolves only finitely many elements of the transcendence base for L. So we see that Fis a rational functions field over a field finitely generated over Qand so it suffices to prove the result for Lfinitely generated over Q. The result is well known in this case (i.e. there are enough discrete valuations). The results above allow us to obtain concrete information about the groups satisfying Max-id for the following types of fields.