Locally nilpotent linear groups with the weak chain conditions on subgroups of infinite central dimension

Publ. Mat. 52 (2008), 151–169 LOCALLY NILPOTENT LINEAR GROUPS WITH THE WEAK CHAIN CONDITIONS ON SUBGROUPS OF INFINITE CENTRAL DIMENSION Leonid A. Kurdachenko, Jos´ e M. Mu˜ noz-Escolano and Javier Otal Abstract Let Vbe a vector space over a field F. If G≤GL(V, F ), the central dimension of Gis the F-dimension of the vector space V/CV(G). In [DEK] and [KS], soluble linear groups in which the set Licd(G) of all proper infinite central dimensional subgroups of Gsatisfies the minimal condition and the maximal condition, respectively, have been described. On the other hand, in [MOS], periodic locally radical linear groups in which Licd(G) satisfies one of the weak chain conditions (the weak minimal condition or the weak maximal condition) have been characterized. In this paper, we begin the study of the non-periodic case by describing locally nilpotent linear groups in which Licd(G) satisfies one of the two weak chain conditions. 1. Introduction Let Vbe a vector space over a field F. The subgroups of the group GL(V, F) of all automorphisms of Vare called linear groups. The Theory of Linear Groups is one of the most developed branches of the Theory of Groups. If Vhas finite dimension over F, it is well-known that GL(V, F ) can be identified with the group of non-singular n×n-matrices with entries in F, where n= dimFV. Finite dimensional linear groups have played an important role in mathematics due to the identification mentioned above and the interplay between other mathematical ideas and such groups. However the study of the subgroups of GL(V, F), 2000 Mathematics Subject Classification. Primary: 20F22; Secondary: 20H20. Key words. Locally nilpotent group, linear group, central dimension of a linear group, weak minimal condition, weak maximal condition. Supported by Proyecto MTM2004-04842 of Direcci´on General de Investigaci´on del Ministerio de Educaci´on y Ciencia (Spain). 152 L. A. Kurdachenko, J. M. Mu˜ noz-Escolano, J. Otal when Vhas infinite dimension over Fhas received considerably less attention, and requires some additional restrictions. In the paper [DEK], a way of studying infinite dimensional linear groups that are near to finite dimensional groups, in some sense, was begun. If Gis a subgroup of GL(V, F ), then Greally acts on the factor-space V/CV(G). Following [DEK], the dimension of this factor-space will be called the central dimension of the subgroup Gand will be denoted by centdimFG. Thus, centdimFG= dimF(V/CV(G)). Suppose that Gis a linear group of finite central dimension. If C=CG(V/CV(G)), then Cis a normal subgroup of Gand G/C is isomorphic to some subgroup of GL(n, F ), where n= dimF(V/CV(G)). Since Cstabilizes the series h0i ≤ CV(G)≤V, Cis abelian. Even more, Cis torsion-free, if charF= 0, and is p-elementary abelian, if char F=p > 0 (see [KW, Proposition 1.C.3] and [FL, Section 43]). Hence, the structure of Gcan be determined by the structure of G/C, which is an ordinary finite dimensional linear group. If G≤GL(V, F), let Licd(G) be the set of all proper subgroups of G of infinite central dimension. In order to study infinite dimensional linear groups Gthat are close to finite dimensional, it is natural to start making Licd(G)very small in some sense. That is, we want to impose some restriction to Licd(G). Given the success that the study of infinite groups with finiteness conditions has enjoyed, it seemed reasonable to study linear groups with finiteness conditions. Thus, in the paper [DEK] linear groups Gin which the set Licd(G) satisfies the minimal condition (or Gsatisfies Min-icd) have been described. The dual case, that is linear groups Gin which Licd(G) satisfies the maximal condition (or Gsatisfies Max-icd) has been studied in the paper [KS]. Since the weak minimal condition and the weak maximal condition are the most natural group-theoretical generalizations of the ordinary minimal and maximal conditions (see [LR, 5.1]), it is natural to go on this direction of research with them. The weak conditions have been introduced by R. Baer [BR] and D. I. Zaitsev [ZD]. We recall their definitions in the most general way. Let Gbe a group and let Mbe a family of subgroups of G. Then Gis said to satisfy the weak minimal condition for M-subgroups, if the set M satisfies the weak minimal condition, that is if given a descending chain of members of M G0≥G1≥ · · · ≥ Gn≥Gn+1 ≥ · · · Locally Nilpotent Linear Groups 153 there exists some m∈Nsuch that the index |Gn:Gn+1|is finite for every n≥m. The definition of the weak maximal condition is dual: Gis said to satisfy the weak maximal condition for M-subgroups, if the set Msatisfies the weak maximal condition, that is if given an ascending chain of members of M G0≤G1≤ · · · ≤ Gn≤Gn+1 ≤ · · · there exists some m∈Nsuch that the index |Gn+1 :Gn|is finite for every n≥m. In passing we note that groups satisfying one of these weak conditions for different Mhave been studied by many authors (see [LR, 5.1] and the surveys [AK], [KK]). We say that a group G≤GL(V, F)satisfies the weak minimal condition on subgroups of infinite central dimension (or Gsatisfies Wmin-icd), if Gsatisfies the weak minimal condition for Licd(G)-subgroups. Dually, we say that G≤GL(V, F )satisfies the weak maximal condition on subgroups of infinite central dimension (or Gsatisfies Wmax-icd), if Gsatisfies the weak maximal condition for Licd(G)-subgroups. Periodic locally radical linear groups satisfying Wmin-icd or Wmax-icd were described in the paper [MOS]: they are either Chernikov or nilpotent-by-abelianby-finite. As a consequence of that description, it was shown that, in the periodic case, the conditions Wmin-icd, Wmax-icd (and even Min-icd) are equivalent. In the current paper, we begin the study of some nonperiodic groups satisfying one of these weak chain conditions as we study locally nilpotent linear groups satisfying Wmin-icd or Wmax-icd. Periodic locally nilpotent linear groups satisfying Wmin-icd or Wmax-icd are Chernikov ([MOS, Corollary D]), but, as we mentioned above, in the non-periodic case, other types of groups can occur, for example minimax groups. The main results obtained of this paper are the following ones. Theorem A (Theorem 2.6).Let Gbe a subgroup of GL(V, F)of infinite central dimension. Suppose that His a normal subgroup of Gsuch that G/H is nilpotent. If Gsatisfies either Wmin-icd or Wmax-icd, then G/H is minimax. In particular, if Gis nilpotent, then Gis minimax. From now on we restrict ourselves to the case of char F=p > 0 since the case of char F= 0 requires a different approach. We recall that, if Gis a locally nilpotent group, then the set t(G) of all elements having finite order is a characteristic subgroup of G. The subgroup t(G) is called the periodic part or the torsion subgroup of the locally nilpotent group G. 154 L. A. Kurdachenko, J. M. Mu˜ noz-Escolano, J. Otal Theorem B (Theorem 3.5).Let Gbe a locally nilpotent subgroup of GL(V, F)of infinite central dimension. Suppose that char F=p > 0. If Gsatisfies either Wmin-icd or Wmax-icd, then G/t(G)is minimax. Let Fbe the class of finite groups. If Gis a group, then the the intersection GFof all subgroups Hof Gof finite index is called the finite residual of G. The factor-group G/GFis said to be residually finite. Obviously, Gitself is residually finite if GF=h1i. Theorem C (Theorem 3.6).Let Gbe a locally nilpotent subgroup of GL(V, F)of infinite central dimension. Suppose that char F=p > 0. If Gsatisfies either Wmin-icd or Wmax-icd, then G/GFis minimax and nilpotent. Let Nbe the class of nilpotent groups. As above the intersection GN of all normal subgroups Hof a group Gsuch that G/H ∈Nis called the nilpotent residual of Gand the factor-group G/GNis said to be residually nilpotent.Gitself is residually nilpotent if GN=h1i. Corollary D (Corollary 3.7).Let Gbe a locally nilpotent subgroup of GL(V, F)of infinite central dimension. Suppose that char F=p > 0. If Gsatisfies either Wmin-icd or Wmax-icd, then G/GNis minimax. 2. Nilpotent linear groups satisfying Wmax-icd or Wmin-icd In this section we establish that nilpotent linear groups (or more precisely, nilpotent factor-groups of linear groups) that satisfy Wmin-icd or Wmax-icd are minimax or have finite central dimension. In order to show this, we need first some preliminary results, the first lemma of which deals with abelian-by-finite linear groups, and it is widely employed throughout this paper. If Gis a group, as usual, we denote by Π(G) the set of primes occurring as divisors of the orders of the periodic elements of G. Lemma 2.1. Let Gbe a subgroup of GL(V, F)of infinite central dimension. Let Hbe a normal subgroup of Gsuch that G/H is abelian-by-finite. If Gsatisfies either Wmin-icd or Wmax-icd, then G/H is minimax. Proof: Suppose that Ghas infinite central dimension and G/H is not minimax. Let Abe a normal subgroup of Gsuch that A/H is abelian and G/A is finite. By our assumption, A/H is not minimax. We note that if Uis some G-invariant subgroup of Awith U≤Hsuch that A/U is periodic, then [MOS, Lemma 4.4] shows that G/U must be Chernikov. In particular, A/U is likewise Chernikov. Locally Nilpotent Linear Groups 155 We claim that the 0-rank (or torsioin-free rank)r0(A/H) of A/H is infinite. For, otherwise, we may choose a finite maximal Z-independent subset {a1H,...,akH}in A/H. Put B/H =ha1H,...,akHiG/H. Since G/A is finite, B/H is a finitely generated abelian subgroup of A/H such that A/B is periodic. Applying the above paragraph, A/B is a Chernikov group. But in this case A/H is minimax, a contradiction. This contradiction shows our claim, that is r0(A/H) is infinite. Let c1Hbe an element of A/H of infinite order and put C1/H = hc1HiG/H. Since G/A is finite, C1/H is a finitely generated G-invariant abelian subgroup of A/H. There is a positive integer tsuch that D1/H = (C1/H)tis free abelian, and, obviously, D1/H is also G-invariant. Suppose that we have already constructed an ascending series h1i=D0/H ≤D1/H ≤ · · · ≤ Dα/H of G-invariant subgroups of A/H whose factors are free abelian. Then the subgroup Dα/H is free abelian (see [KM, Theorem 7.1.3]). If A/Dαis not periodic, then it has an element cα+1Dαof infinite order. Put Cα+1/Dα=hcα+1DαiG/Dα. Since G/A is finite, Cα+1/Dαis a finitely generated G-invariant abelian subgroup of A/Dα. There is an r≥0 such that Dα+1/Dα= (Cα+1/Dα)ris free abelian. By construction, Dα+1/Dαis also G-invariant. Moreover, there exists an ordinal γsuch that A/Dγis periodic. We also remark that E/H =Dγ/H is free abelian. Since A/E is periodic, as we have already noted, A/E is Chernikov. In particular, the set Π(A/E) is finite. Suppose that p /∈ Π(A/E). Since E/H is a free abelian subgroup, E/H 6= (E/H)p=L/H. Furthermore (E/H)/(L/H) is an infinite elementary abelian p-group. It follows that the Sylow p-subgroup of A/L is infinite elementary abelian. If W/L is the Sylow p′-subgroup of A/L, then A/W is an infinite elementary abelian p-group. In particular, it is not Chernikov, which contradicts the fact established above. This final contradiction implies our result. Corollary 2.2. Let Gbe a subgroup of GL(V, F )of infinite central dimension. If Gsatisfies Wmin-icd or Wmax-icd, then G/[G, G]is minimax. The proof of a similar theorem for nilpotency is the final step of this section. In order to do this, we need some results about the existence of certain subgroups of a group Gprovided the first central factor-group of Gsatisfies prescribed properties. 156 L. A. Kurdachenko, J. M. Mu˜ noz-Escolano, J. Otal Lemma 2.3. Let Gbe a group whose center ζ(G)has an infinite elementary abelian p-subgroup Esuch that G/E is abelian minimax. Then Ghas a normal minimax subgroup Lsuch that G=LE. In particular, G/L is an infinite elementary abelian p-group. Proof: Clearly Gis nilpotent. Hence the torsion subgroup T=t(G) of G is characteristic in G. Let Pbe the Sylow p-subgroup of G. Then E≤P. Since G/E is minimax, P/E is a Chernikov subgroup. Let D/E be the divisible part of P/E. Then the abelian group G/D has a finite Sylow p-subgroup P/D and therefore G/D =P/D ×R/D, for some subgroup R(see [FL, Theorem 27.5]). In particular, G/R is a finite p-group. We have D/E =K1/E × · · · × Kr/E, where K1/E, . . . Kr/E are Pr¨ufer p-subgroups. Let 1 ≤j≤r. Since Kj/E is locally cyclic, the inclusion E≤ζ(G) gives that Kjis abelian. By [KL1, Lemma 1], the subgroup Kjhas a Pr¨ufer p-subgroup Vj, so that Kj=EVj. Clearly Vjis a characteristic subgroup of Kj, in particular, it is normal in D. Then U=V1···Vr is a normal divisible Chernikov subgroup of G. Clearly D=EU, and the inclusion E≤ζ(G) implies that Dis abelian. Then EU/U is the Sylow p-subgroup of R/U. Let Q/U be the Sylow p′-subgroup of R/U. We remark that Qis normal in G,Qis minimax and EQ/Q is the periodic part of R/Q. By [HK, Proposition 2], R/Q has a normal torsion-free subgroup W/Q such that R/W is a bounded p-group. Proceeding as in the proof of [HK, Proposition 2], we see that the subgroup W/Q can be chosen such that W/Q is characteristic in R/Q, and hence Wis normal in G. Since W/Q is torsion-free, Wis minimax. The finiteness of G/R implies that G/W is bounded. Hence G/W is a central extension of EW/W by a bounded minimax group G/EW. However a bounded abelian minimax group is finite. In other words, G/W is central-byfinite. It follows that G/W has a normal finite subgroup L/W such that G/W = (L/W )(EW/W ). Clearly Lis likewise minimax. It follows that G/L is an infinite elementary abelian p-group, as required. Locally Nilpotent Linear Groups 157 Corollary 2.4. Let Gbe a group whose center ζ(G)has an infinite elementary abelian p-subgroup Esuch that G/E is nilpotent minimax. Then Ghas a normal subgroup Lsuch that G/L is an infinite elementary abelian p-group. Proof: Let h1i=Z0/E ≤Z1/E ≤ · · · ≤ Zn/E =G/E be the upper central series of G/E. We proceed by induction on n. If n= 1, then G/E is abelian and the statement follows from Lemma 2.3. Suppose that n > 1 and consider the subgroup Z1. By Lemma 2.3, Z1has a normal minimax subgroup Usuch that Z1=UE. Put V= CoreG(U), so that Vis minimax and V≤Z1. Observe that the second center of Gincludes Z1. Therefore for each element zV ∈Z1/V the mapping gV 7→ [gV, zV ] is a homomorphism of G/V into EV/V , whose kernel coincides with CG/V (zV ). Thus [G/V, zV ] is an elementary abelian subgroup. On the other hand, the obvious inclusion EV/V ≤CG/V (zV ) implies that (G/V )/CG/V (zV ) is a nilpotent minimax group. The isomorphism [G/V, zV ]∼ =(G/V )/CG/V (zV ) shows that the last factor-group is a bounded minimax group, in particular, it is finite. This means that Z1/V ≤F C(G/V ), where FC(G/V ) is the FC-center of G/V . Now let W/V = (U/V )G/V , then W/V is finite and hence Wis minimax. Therefore Z1=WE, where Wis a minimax G-invariant subgroup. In particular, Z1/W is an infinite elementary abelian p-group, Z1/W ≤ζ(G/W) and (G/W)/(Z1/W ) is a nilpotent minimax group whose nilpotency class is n−1. Consequently we can apply now the inductive hypothesis. Lemma 2.5. Let Gbe a group and suppose that ζ(G)has a divisible abelian p-subgroup Esuch that Eis not Chernikov and G/E is nilpotent minimax. Then Ghas a normal minimax subgroup Lsuch that G/L is a divisible abelian p-group which is not Chernikov. Proof: Let h1i=Z0/E ≤Z1/E ≤ · · · ≤ Zn/E =G/E be the upper central series of G/E. Clearly, Gis nilpotent. Hence the set T=t(Z1) of all elements of Z1having finite order is a characteristic subgroup of Z1. Let Pbe the Sylow p-subgroup of Z1. Then E≤P. Since G/E is minimax, P/E is a Chernikov subgroup. Let D/E be the 158 L. A. Kurdachenko, J. M. Mu˜ noz-Escolano, J. Otal divisible part of P/E. Therefore Dis periodic nilpotent divisible, so that it is abelian (see [KA,§65]). Let Qbe the Sylow p′-subgroup of T. Remark that Qis normal in Gand Qis Chernikov. For each element gQ ∈Z1/Q the mapping xQ 7→ [gQ, xQ], x∈G, is a homomorphism of G/Q into DQ/Q, whose kernel coincides with CG/Q(gQ). On the other hand, the obvious inclusion EQ/Q ≤CG/Q(gQ) implies that (G/Q)/CG/Q(gQ) is a nilpotent minimax group. The isomorphism [gQ, G/Q]∼ =(G/Q)/CG/Q(gQ) shows that the latter is a periodic minimax group and hence Chernikov. Since Z1/D is abelian minimax, it has a finitely generated subgroup F/D such that Z1/F is periodic. Let F/D =hg1D,...,gtDi. As we have seen above, the subgroups [g1Q, G/Q],...,[gtQ, G/Q] are Chernikov. Put U/Q = [g1Q, G/Q]···[gtQ, G/Q]. Then Uis a Chernikov subgroup and V/U =hg1,...,gtiU/U ≤ζ(G/U). In particular; Vis normal in Gand V/U is a finitely generated abelian group. Furthermore, Vis minimax, so that DV/V is not Chernikov. Since the factor-group Z1/V is periodic nilpotent, Z1/V =P1/V ×Q1/V, where P1/V is the Sylow p-subgroup of Z1/V and Q1/V is the Sylow p′-subgroup of Z1/V . Clearly Q1is minimax and Z1/Q1is a p-group. Since DQ1/Q1is divisible and not Chernikov and Z1/DQ1is Chernikov, Z1/Q1has a normal divisible subgroup D1/Q1of finite index which is not Chernikov. Recall that in a nilpotent periodic group every divisible subgroup lies in the center (see [KW, 1.F.1]), so that Z1/Q1is central-by-finite. As above we see that Z1/Q1has a G-invariant Chernikov subgroup L/Q1such that Z1/Q1= (L/Q1)(D1/Q1). Clearly Lis likewise minimax. It follows that Z1/L is a divisible abelian p-group which is not Chernikov. Proceeding in the same way, after finitely many steps we obtain the required result. Before stating the last result of the section, we recall some definitions and facts that we use subsequently. Let Abe an abelian group of finite special rank, Ma maximal Z-independent subset of Aand set B=hMi. Locally Nilpotent Linear Groups 159 Put Sp(A) = {p|pis a prime such that the Sylow p-subgroup of A/B is infinite}. The set Sp(A) is called the spectrum of the group A. If Vis also a finitely generated subgroup of Asuch that A/V is periodic, then B/(B∩V)∼ =BV/V and V/(B∩V)∼ =BV/B are finite, which shows that the set Sp(A) is independent of the choice of the finitely generated subgroup B. Let Gbe a nilpotent group of finite special rank and let h1i=ζ0≤ζ1≤ · · · ≤ ζn=G be the upper central series of G. Put Sp(G) = Sp(ζ1/ζ0)∪ · · · ∪ Sp(ζn/ζn−1). It is not hard to see that the spectrum of Gis the union of the spectrum of all factors of every central series of G. We note that, if His a normal subgroup of Gsuch that G/H is periodic and pis a prime such that p /∈Sp(G), then the Sylow p-subgroup of G/H is finite. We are now in a position to establish Theorem A, the main result of this section. Theorem 2.6. Let Gbe a subgroup of GL(V, F )of infinite central dimension. Suppose that His a normal subgroup of Gsuch that G/H is nilpotent. If Gsatisfies either Wmin-icd or Wmax-icd, then G/H is minimax. Proof: By hypothesis, Ghas infinite central dimension. Let h1i=Z0/H ≤Z1/H ≤ · · · ≤ Zn/H =G/H be the upper central series of G/H. We proceed by induction on n. If n= 1, then G/H is abelian and the statement follows from Corollary 2.2. Suppose that n > 1 and we have already proved that G/Z1is minimax. We want to prove that Z1/H is minimax. Suppose the contrary and seek a contradiction. It suffices to show that Ghas a normal subgroup Usuch that G/U is periodic abelian and not minimax. Then Corollary 2.2 will give the required contradiction. Put L=G/H,C0=Z0/H,...,Cn=Zn/H. Then C1≤ζ(L). Choose in C1a maximal Z-independent subset {bλ|λ∈Λ}and put B=hbλ|λ∈Λi= Drλ∈Λhbλi, so that C1/B is a periodic abelian group. 166 L. A. Kurdachenko, J. M. Mu˜ noz-Escolano, J. Otal Put L(q) = T{H|H∈M(q)}. By [KNKL, Lemma 2], G/L(q) is nilpotent and t(G/L(q)) is finite. Also we note that, by [MOS, Corollary 2.7], the factor-group G/[G, G]Gqhas to be finite. Let Q/L be the Sylow q-subgroup of G/L. If 1 6=xL ∈Q/L, then there exists a subgroup H∈Msuch that xL 6∈ H/L. Let U/H be the Sylow p′-subgroup of G/H. Then G/U is a finite q-group and x /∈U. In other words, for every element xL ∈Q/L there is a subgroup Ux∈M(q) such that xL /∈Ux/L. It follows that Q∩L(q) = L, that is Q/L ∼ =Q/(Q∩L(q)) ∼ =QL(q)/L(q). Therefore the finiteness of t(G/L(q)) implies that Q/L is finite. This yields that Q/L has a maximal G-invariant subgroup R(q)/L. Since Gis locally nilpotent, Q/R(q) is a G-central factor (see [KA,§63]). Let T/L be the periodic part of G/L and put R= Drq∈Π(G/L)R(q). Then T/R ≤ζ(G/R). By Theorem 3.5 G/T is minimax, so that it has finite special rank. Applying [MN], G/T is nilpotent and consequently so too is G/R. By Theorem 2.6, G/R is minimax. In this case T/R is finite. Therefore Π(G/L) is finite. Since every Sylow q-subgroup of G/L is finite, T/L is finite. Then G/L is nilpotent, and applying Theorem 2.6 again, we conclude that G/L is minimax. Corollary 3.7. Let Gbe a locally nilpotent subgroup of GL(V, F)of infinite central dimension. Suppose that char F=p > 0. If Gsatisfies either Wmin-icd or Wmax-icd, then G/GNis minimax. Proof: Let Mbe the family of all normal subgroups Hsuch that G/H is nilpotent, and let Lbe the nilpotent residual of G, so that L=T{M| M∈M}. By Theorem 3.5, G/t(G) is minimax, and, in particular, G/t(G) has finite special rank. We recall that torsion-free locally nilpotent groups of finite special rank are nilpotent ([MN]); whence t(G)∈ M. Therefore L≤t(G). Put T=t(G). If Thas infinite central dimension, then Tis a Chernikov subgroup by [MOS, Corollary D]. If Thas finite central dimension, then, by Proposition 3.1, the Sylow p-subgroup Qof T is bounded nilpotent and T/Q is abelian-by-finite Chernikov. Hence (G/L)/(QL/L) is minimax and QL/L is bounded nilpotent. Let H∈M, then G/H is minimax by Theorem 2.6. Since a bounded minimax group is finite, the isomorphisms (QL/L)/((H/L)∩(QL/L)) ∼ =QL/(H∩QL)∼ =QH/H ≤G/H Locally Nilpotent Linear Groups 167 show that (QL/L)/((H/L)∩(QL/L)) is finite. Put U/L = (H/L)∩ (QL/L). We have T/L =QL/L ×R/L, where R/L is the Sylow p′-subgroup of T/L. Put W/L = (U/L)(R/L). It follows that (T/L)/(W/L) is a finite p-group. Since G/T is nilpotent and T/W is finite, G/W is nilpotent too. Then it has a torsion-free normal subgroup D/W such that G/D is bounded ([HK, Proposition 2]). Since G/W has finite special rank, G/D, being bounded, is finite. Furthermore, D/W is torsion-free, so (T/L)∩(D/L) = W/L. The equation L=TMshows that for each element xL ∈QL/L there is a normal subgroup W/L of finite index such that xL /∈W/L. It follows that (QL/L)∩(GF/L) = h1i. In other words, QL/L is isomorphic to some subgroup of G/GF. By Theorem 3.6, G/GFis minimax. Thus QL/L is finite. Since (G/L)/(QL/L) is minimax, G/L is likewise minimax. References [AK] O. D. Artemovych and L. A. 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An account of the group-theoretic properties of infinite groups of matrices”, Ergebnisse der Matematik und ihrer Grenzgebiete 76, SpringerVerlag, New York-Heidelberg, 1973. [ZD] D. I. Zaitsev, Groups satisfying the weak minimal condition, Ukrainian Math. J. 20(4) (1968), 408–416. Locally Nilpotent Linear Groups 169 Leonid A. Kurdachenko: Department of Algebra University of Dnepropetrovsk Vul. Naukova 13 Dnepropetrovsk 50 Ukraine 49050 E-mail address:lkurdachen[email protected] Jos´e M. Mu˜noz-Escolano and Javier Otal: Department of Mathematics University of Zaragoza Pedro Cerbuna 12 50009 Zaragoza Spain E-mail address:[email protected] E-mail address:[email protected] Primera versi´o rebuda el 20 de novembre de 2006, darrera versi´o rebuda el 12 de juny de 2007.