View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 208 (2007) 785–795 www.elsevier.com/locate/jpaa Linear groups with rank restrictions on the subgroups of infinite central dimension Olga Yu Dashkovaa, Martyn R. Dixonb,∗, Leonid A. Kurdachenkoa a Department of Algebra, University of Dnepropetrovsk, Vulycya Naukova 13, Dnepropetrovsk 50, 49050, Ukraine b Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, USA Received 2 December 2005; received in revised form 2 March 2006 Available online 5 May 2006 Communicated by S. Donkin To Hermann Heineken, on his 70th birthday Abstract The authors study linear groups of infinite central dimension and of infinite p-rank all of whose proper subgroups of infinite p-rank are of finite central dimension. c 2006 Elsevier B.V. All rights reserved. MSC: Primary: 20F19; secondary: 20H20; 20E15 1. Introduction A group G that is isomorphic to a group of automorphisms of a vector space A over a field F is called a linear group and we denote the group of all such automorphisms by GL(F, A). If dimF A, the dimension of A over F, is finite, then G is usually called a finite dimensional linear group and GL(F, A) can then be identified with the group of n × n matrices with entries in F, where n = dimF A. Finite dimensional linear groups have played an important role in mathematics, partly because of this correspondence, but also because of the rich interplay between geometrical and algebraic ideas associated with such groups. The study of the subgroups of GL(F, A) in the case when A is infinite dimensional over F has been much more limited and normally requires some additional restrictions. One such restriction that has enjoyed considerable attention is the notion of a finitary linear group, where a group G is called finitary if, for each element g ∈ G, the subspace C A(g) has finite codimension in A, and the reader is referred to the paper [11] and the survey [12] to see the type of results that have been obtained. Given the success that the study of groups with finiteness conditions has enjoyed, it is also reasonable to study infinite dimensional linear groups with finiteness conditions. From our point of view a finitary linear group can be viewed as the linear analogue of an FC-group (that is, a group with finite conjugacy classes). More ∗ Corresponding author. E-mail addresses: [email protected] (M.R. Dixon), [email protected], [email protected] (L.A. Kurdachenko). 0022-4049/$ - see front matter c 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jpaa.2006.04.002 786 O. Yu Dashkova et al. / Journal of Pure and Applied Algebra 208 (2007) 785–795 recently, papers [2] and [8] have discussed certain other types of finiteness condition in linear groups, analogous to the well-known theories of groups with the minimal condition (respectively maximal condition), first investigated by S.N. Cernikovˇ (respectively K.A. Hirsch). If H is a subgroup of GL(F, A) then H really acts on the quotient space A/C A(H) in a natural way. We denote dimF (A/C A(H)) by centdim F H. If dimF (A/C A(H)) is finite we say that H has finite central dimension; otherwise we shall say that H has infinite central dimension. A group G is therefore finitary linear precisely when each of its cyclic subgroups has finite central dimension. (We remark that the central dimension of a subgroup depends on the particular vector space on which it is acting.) If H ≤ GL(F, A) has finite central dimension then it is easy to see that if K ≤ H then K also has finite central dimension. In paper [2] we studied linear groups in which the family of subgroups of infinite central dimension satisfied the minimal condition; paper [8] considered the analogous problem for linear groups in which the family of subgroups of infinite central dimension satisfied the maximal condition. In this paper we take this study a stage further and consider linear groups of infinite central dimension satisfying certain restrictions on the ranks of their subgroups. A group G is said to have finite 0-rank (or finite torsion-free rank), r0(G) = r, if G has a finite subnormal series with exactly r infinite cyclic factors, all other factors being periodic. It is well known that the 0-rank is independent of the chosen series. In the case of polycyclic-by-finite groups the 0-rank is simply the Hirsch number. It is also well known that if G is a group of finite 0-rank, H ≤ G and L C G, then H and G/L also have finite 0-rank. Furthermore r0(H) ≤ r0(G) and r0(G) = r0(L) +r0(G/L). Soluble groups of finite torsion-free rank have been well studied (see [9, Section 5.2], for example). This theory has been taken somewhat further in the paper [3] where even locally (soluble-by-finite) groups of finite torsion-free rank are considered. Now let p be a prime. A group G has finite p-rank, r p(G) = r, if every elementary abelian p-section of G is finite of order at most pr and there is an elementary abelian p-section U/V such that |U/V | = pr . It is well known that if G is a group of finite p-rank, H ≤ G and L C G, then H and G/L also have finite p-rank. Furthermore r p(H) ≤ r p(G) and r p(G) ≤ r p(L) + r p(G/L). In our work when speaking of the p-rank we shall assume that p = 0 or that p is a prime, being more specific when we need to be. We are interested in those linear groups of infinite central dimension and of infinite p-rank whose proper subgroups of infinite p-rank are of finite central dimension; thus the proper subgroups of infinite p-rank, although large, are not too large, in some sense. In Section 2 we cover the basic results concerning groups of infinite central dimension and infinite p-rank whose proper subgroups of infinite p-rank are of finite central dimension. Given the definition of groups with finite 0-rank, it makes sense to add some additional solubility hypothesis and this we do in Section 3, where we obtain our first main result on soluble groups of the type under discussion. It turns out that when p = 0 there are no groups of the type in question. When p 6= 0 there is additional work to be done. We can also obtain other results with a little more effort. Accordingly, we recall that a group G has finite abelian section rank if every abelian section of G has finite p-rank for p ≥ 0. We note that Baer and Heineken [1] have shown that for soluble (and even hyperabelian) groups finite abelian section rank is equivalent to finite abelian subgroup rank (that is, the abelian subgroups of G have finite p-rank for p ≥ 0). Furthermore, a group G has finite special rank r(G) = r if every finitely generated subgroup of G can be generated by r elements and r is the least positive integer with this property. This notion is due to Mal’cev [10]. The special rank of a group is sometimes also called the Prufer–Mal’cev¨ rank. In Section 4 we obtain results aimed at generalizing our work to the locally soluble case. Our notation will be as follows. We let X0 denote the class of groups G such that r0(G) is finite and, if p is a prime, let X1,p denote the class of groups G for which r p(G) is finite. At the risk of some ambiguity, we will denote X1,p by X1. We let X2 denote the class of groups with finite abelian section rank and we let X3 denote the class of groups that have finite special rank. In Section 4 we prove the following omnibus theorem, where in the case i = 1 the prime p occurring in the statement of the theorem is the same prime as was used in the definition of X1. Main Theorem. Let i be one of 0, 1, 2 and 3. Let G ≤ GL(F, A) be locally soluble and suppose that centdim F (G) is infinite and G 6∈ Xi . Suppose that every proper subgroup of G has finite central dimension or belongs to the corresponding class Xi . Then G is soluble and if i = 1, 2 or 3 the following conditions hold: ∼ (i) G = H o Q where Q = Cq∞ for some prime q; (ii) H is a p-group of finite central dimension where p = char F 6= q; (iii) K = H ∩ Z(G) is finite; O. Yu Dashkova et al. / Journal of Pure and Applied Algebra 208 (2007) 785–795 787 (iv) H/K is an infinite elementary abelian p-group; (v) H/K is a minimal normal subgroup of G/K. In our brief Section 5 we give an example to illustrate our main theorems. Notation that is used but not explained in the sequel is that in standard use and we refer the reader to [13] for this. 2. Preliminary results In this section we shall establish some of the elementary properties of groups in which every proper subgroup of infinite p-rank has finite central dimension.