Groups with finiteness conditions on some subgroup systems: a contemporary stage

This paper gives a brief historical survey of results in which certain systems of subgroups of a group satisfy various finiteness conditions.

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spelling	Dixon, M.R. Subbotin, I.Ya. 2019-06-15T16:52:59Z 2019-06-15T16:52:59Z 2009 Groups with finiteness conditions on some subgroup systems: a contemporary stage / M.R. Dixon, I.Ya. Subbotin // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 4. — С. 29–54. — Бібліогр.: 91 назв. — англ. 1726-3255 https://nasplib.isofts.kiev.ua/handle/123456789/154615 This paper gives a brief historical survey of results in which certain systems of subgroups of a group satisfy various finiteness conditions. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Groups with finiteness conditions on some subgroup systems: a contemporary stage Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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title	Groups with finiteness conditions on some subgroup systems: a contemporary stage
spellingShingle	Groups with finiteness conditions on some subgroup systems: a contemporary stage Dixon, M.R. Subbotin, I.Ya.
title_short	Groups with finiteness conditions on some subgroup systems: a contemporary stage
title_full	Groups with finiteness conditions on some subgroup systems: a contemporary stage
title_fullStr	Groups with finiteness conditions on some subgroup systems: a contemporary stage
title_full_unstemmed	Groups with finiteness conditions on some subgroup systems: a contemporary stage
title_sort	groups with finiteness conditions on some subgroup systems: a contemporary stage
author	Dixon, M.R. Subbotin, I.Ya.
author_facet	Dixon, M.R. Subbotin, I.Ya.
publishDate	2009
language	English
container_title	Algebra and Discrete Mathematics
publisher	Інститут прикладної математики і механіки НАН України
format	Article
description	This paper gives a brief historical survey of results in which certain systems of subgroups of a group satisfy various finiteness conditions.
issn	1726-3255
url	https://nasplib.isofts.kiev.ua/handle/123456789/154615
citation_txt	Groups with finiteness conditions on some subgroup systems: a contemporary stage / M.R. Dixon, I.Ya. Subbotin // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 4. — С. 29–54. — Бібліогр.: 91 назв. — англ.
work_keys_str_mv	AT dixonmr groupswithfinitenessconditionsonsomesubgroupsystemsacontemporarystage AT subbotiniya groupswithfinitenessconditionsonsomesubgroupsystemsacontemporarystage
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fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics SURVEY ARTICLE Number 4. (2009). pp. 29 – 54 c⃝ Journal “Algebra and Discrete Mathematics” Groups with finiteness conditions on some subgroup systems: a contemporary stage Martyn R. Dixon, Igor Ya. Subbotin Communicated by V. V. Kirichenko Dedicated to Leonid Kurdachenko, with heartfelt thanks for his friendship, on the occasion of his 60th birthday Abstract. This paper gives a brief historical survey of results in which certain systems of subgroups of a group satisfy various finiteness conditions. 1. Introduction One of the main themes in group theory (finite or infinite) is the study of the influence of systems of subgroups on the structure of a group. The structure of a group depends, to a significant extent, on the presence of a system of subgroups with certain properties, the size of this system, and the interaction of this system with other subgroups. There is a wide variety of cases that can be studied. Sometimes the presence of a single subgroup with given properties can be very influential on the structure of a group whereas, in other cases, a group can have many subgroups with some given property, but the influence of this system of subgroups need not be significant. There are two typical subgroup properties that are often used–properties that are internal to the group such as the properties of being a normal or subnormal subgroup and properties that are external to the group such as the property of belonging to some class X of groups. The choices for X include the classes of abelian groups and the class of nilpotent groups, to name but two. A natural approach to studying groups was established. In this ap- proach a group theoretical property P was chosen and groups in which the system of P-subgroups, LP(G), was quite large were considered. For Jo u rn al A lg eb ra D is cr et e M at h .30 Finiteness conditions on some subgroup systems example, LP(G) can be taken to coincide with the family of all (proper) subgroups of G. As techniques became established the size of the system LP(G) could be gradually reduced so that the system Lnon−P(G) of all non–P–subgroups grew larger. The first step in this program was taken by R. Dedekind in his classical paper [16] where he described the finite groups, all of whose subgroups are normal. This is precisely the class of finite groups G in which the system Lnorm(G), consisting of all normal subgroups, coincides with the system of all subgroups (or in which the system Lnon−norm(G) is empty). In their famous paper [63], G. Miller and H. Moreno described the finite groups all of whose proper subgroups are abelian; this class coincides with the class of finite groups G where Lab(G), the system of all abelian subgroups, coincides with the family of all proper subgroups. Alternatively, this class is the class of finite groups in which the family Lnon−ab(G) of all non–abelian subgroups consists of at most one element, the group G. In this setting, we need to mention the remarkable article [73] due to O. Yu. Schmidt, which completely describes the finite groups all of whose proper subgroups are nilpotent. O. Yu. Schmidt continued Dedekind’s research in the paper [74] where he described the finite groups G in which the subgroups in the family Lnon−norm(G) are conjugate. In [75] he also completely described the finite groups in which Lnon−norm(G) is the union of two conjugacy classes. Many authors have continued these investigations in both finite and infinite groups. The situation is especially promising, but challenging, in infinite group theory. There is a large variety of situations where the con- cepts “to be quite small” and “to be very large” can be studied with fruit- ful results. S. N. Chernikov introduced one such effective approach. He began the investigation of groups G where Lnon−P(G) satisfies some nat- ural finiteness condition. Such finiteness conditions include, in particular, such classical finiteness conditions as the minimal and the maximal con- ditions. In his work [7], he studied groups in which the family Lnon−ab(G) satisfies the minimal condition and, in the paper [8], he considered groups G in which the set Lnon−norm(G) consists of finite subgroups only. The main goal of this article is to survey some important developments in the above-mentioned area that have been achieved over the last few decades. This is a huge area of research. Our choice of what to include and that to omit has been guided by our own interests and undoubtedly there are many interesting results that will not be mentioned here. For this reason we have decided to frame our discussion in the context of the most important properties such as normality, subnormality, almost normality, commutativity and their generalizations. Jo u rn al A lg eb ra D is cr et e M at h .M. R. Dixon, I. Ya. Subbotin 31 2. Groups with “small” systems of non-normal subgroups As we mentioned above, the finite groups with all subgroups normal were described by R. Dedekind [16]. Later, R. Baer [2] extended Dedekind’s results to infinite groups and the groups obtained have subsequently been termed Dedekind groups. These groups have a very simple structure, namely, a Dedekind group either is abelian or is a group of the type A × B × Q where A is an abelian periodic 2́-group, B is an elementary abelian 2-group, and Q is a quaternion group. In extending these results, S. N. Chernikov [8] considered groups all of whose infinite subgroups are normal, those groups G in which the system Lnon−norm(G) of all non- normal subgroups consists of only finite subgroups. Naturally in infinite groups one would expect the structure of the infinite subgroups to play a dominant role. This class of groups is wider then the class of Dedekind groups and S. N. Chernikov obtained the following description of such groups [8]. Theorem 2.1. Let G be an infinite group all of whose infinite subgroups are normal. (i) If G is non-abelian, then G is periodic; (ii) If G is locally finite then G is either Dedekind, or G contains a normal Prüfer subgroup K such that G/K is a finite Dedekind group. The latter result here was obtained by S. N. Chernikov, not just for locally finite groups, but for periodic groups with the additional assump- tion that the group G has an infinite abelian subgroup. It is a well-known theorem of P. Hall and C. Kulatilaka [28] that every infinite locally finite group contains an infinite abelian subgroup. Sophisticated examples of infinite groups all of whose subgroups are finite have been constructed by A. Yu. Olshanskii [68, § 28], so that some condition such as G being locally finite in Theorem 2.1 is required. Groups in which Lnon−norm(G) satisfies the minimal condition (groups with the condition Min− (non−norm)) have also been studied by S. N. Chernikov [12] where the following results are obtained. Theorem 2.2. Let G be an infinite group satisfying the condition Min− (non− norm). (i) If G is not periodic, then G is abelian. (ii) If G is locally finite, then G is either Dedekind or Chernikov. Jo u rn al A lg eb ra D is cr et e M at h .32 Finiteness conditions on some subgroup systems We note that some generalizations of these results have been obtained in the articles [8, 12], but we refrain from discussing these generalizations since they have been described in other surveys (see, [9, 10, 11, 90, 6]). We should also point out the paper [70] where the hypotheses are weakened even further. The maximal condition is dual to the minimal condition. The groups G in which Lnon−norm(G) satisfies the maximal condition (the Max − (non−norm) condition) were studied in the articles of L. A. Kurdachenko, N. F. Kuzennyi and N. N. Semko [37] and G. Cutolo [13]. We note, that although the class of locally graded groups with the condition Min − (non−norm) is the union of the classes of Dedekind groups and Chernikov groups, the situation is very different for the class of groups with the con- dition Max − (non − norm); here as usual a group is locally graded if every nontrivial finitely generated subgroup has a nontrivial finite image. The main results of the above mentioned articles are captured by the following result. Theorem 2.3. Let G be a locally graded group with the condition Max− (non− norm). Then G is a group of one of the following types. (i) G is an almost polycyclic group; (ii) G is a Dedekind group; (iii) �(G) contains a Prüfer p−subgroup P such that G/P is a finitely generated Dedekind group; (iv) G = H × L where H ∼= Q2 and L is a finite non-abelian Dedekind group. Here, Qp denotes the additive group of rationals of the form apb where p is a prime and a, b are integers. Next we note that if all finitely generated subgroups of a group G are normal, then all subgroups of G are normal in G. By contrast, groups all of whose infinitely generated subgroups are normal are more complicated, where here, by infinitely generated we mean not finitely generated. In this case the system Lnon−norm(G) consists of only finitely generated subgroups. Such groups have been studied in the articles of L. A. Kurdachenko and V. V. Pylaev [41], G. Cutolo [13], and G. Cutolo and L.A. Kurdachenko [14]. The main results of these papers can be formulated in the following way. Theorem 2.4. Let G be a group that has an ascending series of sub- groups whose factors are locally (soluble-by-finite). Every infinitely gen- erated subgroup of G is normal if and only if G is a group of one of the following types: Jo u rn al A lg eb ra D is cr et e M at h .M. R. Dixon, I. Ya. Subbotin 33 (1) G is a Dedekind group; (2) G contains a normal Prüfer subgroup K such that G/K is a finitely generated Dedekind group; (3) G satisfies the following conditions: (3a) the center �(G) contains a Prüfer p-subgroup K such that G/K is a minimax abelian group with finite periodic part; (3b) Sp(G/K) = {p}; (3c) G/FC(G) is torsion-free; (3d) if A is an abelian subgroup in G, then A/(A ∩ K) is finitely generated; (4) G = T ×A, where A ∼= Q2 and T is a finite Dedekind group; (5) G satisfies the following conditions: (5a) G = (A × T ) ⋊ ⟨g⟩ where A ∼= Qp for some prime p and T is a finite Dedekind group; (5b) if T is non-abelian, then p = 2; (5c) the element g induces a power automorphism on the Sylow p-subgroup Tp of the group T ; (5d) there exists r ∈ ℕ such that ag = ac where c = pr or c = −pr for each a ∈ ATp′ (where Tp′ is a Sylow p′-subgroup of T ). In the articles [3] and [88], new interesting finiteness conditions were introduced, namely the weak minimal and weak maximal conditions for different types of subgroups. These conditions have proved to be very successful, stimulating a lot of research in this area. The results of the published research have been reflected in the survey [34] and so we here consider only the results of that research which are relevant to us here. Let M be some system of subgroups of a group G. We will say that M satisfies the weak minimal condition (respectively the weak maximal condition) or that G satisfies the weak minimal condition (respectively the weak maximal condition) on M-subgroups (which we write as Min-∞-M or Max-∞-M respectively) if G has no infinite descending (respectively ascending) chain {Hn ∣ n ∈ ℕ} of M−subgroups such that the indices ∣Hn : Hn+1∣ (respectively ∣Hn+1 : Hn∣) are infinite for every n ∈ ℕ. If M = Lnon−norm(G), then we obtain groups with the weak mini- mal condition (respectively the weak maximal condition) on non-normal subgroups which we denote by Min-∞-(non-norm) (respectively Max- ∞-(non-norm)). The structure of such groups was described by L. A. Jo u rn al A lg eb ra D is cr et e M at h .34 Finiteness conditions on some subgroup systems Kurdachenko and V. E. Goretskii in [36], where the following result was obtained. Theorem 2.5. A locally (soluble-by-finite) group G satisfies the condi- tion Min-∞-(non-norm) (respectively Max-∞-(non-norm)) if and only if G either is a Dedekind group or a minimax group. A large number of papers have been devoted to the study of groups with restriction on the system Lnon−norm(G). Certainly, if all cyclic sub- group of a group are normal, then all subgroups of this group are normal which makes it natural to consider those groups in which Lnon−norm(G) consists of only cyclic subgroups. This leads to the study of groups in which all non-cyclic subgroups are normal, a class of groups first studied by S. N. Chernikov [9]. The papers [55]–[59] due to F. N. Liman have also been dedicated to this question. More generally, the works [65, 61], [49]– [54], [71, 72, 76, 78] have been dedicated to the so called metahamiltonian groups–those groups in which Lnon−norm(G) consists of only abelian sub- groups. Finite metahamiltonian groups were described in [65, 61] and a complete description of metahamiltonian groups was obtained in the work due to N. F Kuzennyi and N. N. Semko [49]–[54],[76]. A natural continuation of these investigations is the exploration of the case when all subgroups of Lnon−norm(G) belong to some class which is a natural extension of the class of abelian groups. Thus, in [38, 39], groups have been studied in which the subgroups in the system Lnon−norm(G) all have finite derived subgroup (respectively, are FC-groups). The main results of these papers can be summarized as follows. Theorem 2.6. (i) Let G be a non–nilpotent group with Chernikov derived subgroup K and let D be the divisible part of K. If every non–FC–subgroup of G is normal in G and CG(D) ∕= G, then G = DL where L is a subgroup having finite derived subgroup and D∩L is a finite G–invariant subgroup. Moreover, every non–normal subgroup of G has finite derived subgroup. (ii) Let G be an almost FC−group. If every non–FC–subgroup of G is normal, then either G is an FC−subgroup or [G,G] is a Chernikov group. (iii) Let G be a metabelian group whose non–normal subgroups are FC– groups. If G is not locally nilpotent, then either G is an FC–group or [G,G] is a Chernikov group. (iv) Let G be a metabelian locally nilpotent group whose non–normal subgroups are FC–groups. Then either G is a Fitting group or [G,G] is a Chernikov group. Jo u rn al A lg eb ra D is cr et e M at h .M. R. Dixon, I. Ya. Subbotin 35 (v) Let G be a group whose non–normal subgroups are FC–groups and let F be the FC–center of G. Then either G is an almost FC–group or F is nilpotent with nilpotency class at most 3 and has a normal subgroup H such that �(F ) ≤ H,H/�(F ) is abelian and F/H is a quaternion group. (vi) Let G be a soluble–by–finite group whose non–normal subgroups are FC–groups. If G is not an FC–group and G/[G,G] is finitely generated, then [G,G] is a Chernikov group. (vii) Let G be a soluble–by–finite group whose non–normal subgroups are FC–groups. If G/[G,G] is periodic divisible then G is an almost FC–group. In particular, either G is an FC–group or [G,G] is a Chernikov group. 3. Groups with “small” systems of non-(almost normal) subgroups A subgroup H of a group G is called almost normal in G if the set of all conjugates of H, clG(H) = {Hg ∣ g ∈ G}, is finite. If a subgroup H is normal in G, then clG(H) = {H}; thus almost normality is a natural generalization of normality. The subgroup H is almost normal in G if and only if its normalizer NG(H) has finite index in G, which gives a good justification for the name of these subgroups. It is clear that the intersection of two almost normal subgroups is almost normal and that a subgroup generated by two almost normal subgroups is almost normal. Thus the set Lan(G) of all almost normal subgroups is a lattice but, in contrast to the lattice of normal subgroups, the lattice of almost normal subgroups is not complete. The groups G for which Lan(G) is complete have been considered by L. A. Kurdachenko and S. Rinauro in [42]. There they proved that it is often the case that such groups are central-by-finite, which means that they have a central subgroup of finite index. The central-by-finite groups here play a role similar to the role that Dedekind groups play in the study of groups G in which Lnon−norm(G) is “small”. Two classical results illustrate this very well. If G is a group all of whose subgroups are almost normal then a well-known theorem of B. H. Neumann [66] asserts that G is central-by-finite. Clearly the hypothesis on G here is equivalent to Lnon−an(G) = ∅. Also, if G is a group all of whose abelian subgroups are almost normal then a theorem of I. I. Eremin [23] implies that G is central-by-finite and clearly in this case the system Lnon−an(G) consists of non-abelian subgroups. I. I. Eremin began the study of groups in which the set Lnon−an(G) is small by considering the case when it consists of finite subgroups [24] Jo u rn al A lg eb ra D is cr et e M at h .36 Finiteness conditions on some subgroup systems and he obtained some conditions under which these groups are central-by- finite. Later L. A. Kurdachenko, S. S. Levishenko and N. N. Semko [77] described the locally (soluble-by-finite) such groups and we give this de- scription next. Theorem 3.1. Let G be an infinite locally (soluble-by-finite) group. (I) If G is non-periodic, then each infinite subgroup of G is almost normal if and only if G is a group of one of the following types: (Ia) G is central-by-finite; (Ib) G = A ⋊ ⟨b⟩, where ∣b∣ = p, for some prime p, A = CG(A) is free abelian of 0-rank p− 1 and b induces a rationally irre- ducible automorphism on A (so every non-trivial ⟨b⟩-invariant subgroup of A has finite index in A); (Ic) G contains a finite normal subgroup F such that G/F is a group of type (Ib). (II) If G is periodic, then every infinite subgroup of G is almost normal if and only if G is a group of one of the following types: (IIa) G is central-by-finite; (IIb) G = D⋊⟨g⟩, where D = CG(D) is a divisible abelian subgroup of special rank p − 1, p is a prime, gp ∈ D and every proper ⟨g⟩-invariant subgroup of D is finite; (IIc) G = D⋊⟨g⟩, where D = CG(D) is a divisible abelian subgroup of special rank at most q − 1, q is the smallest prime in the set Π(⟨g⟩) = {∣g∣ : g is a p′-element} and for every element 1 ∕= y ∈ ⟨g⟩ every proper ⟨y⟩-invariant subgroup of D is finite; (IId) G contains a finite normal subgroup F that G/F is a group of the types (IIb) or (IIc). These results were generalized by S. Franciosi, F. de Giovanni and L. A. Kurdachenko in [26] to the case when the set Lnon−an(G) consists of finitely generated subgroups. For example they showed that if G is a group with an ascending series of subgroups, every factor of which is either locally nilpotent or finite, then if every infinitely generated subgroup of G is almost normal either G/�(G) is finite, or G is a soluble A3-group. Almost soluble A3 -groups in which the set Lnon−an(G) consists of finitely generated subgroups occur in several different families that have been described in detail in [26]. The question of S. N. Chernikov con- cerning the structure of a group G in which the set Lnon−an(G) consists of non-cyclic subgroups was also considered in [26]. Jo u rn al A lg eb ra D is cr et e M at h .M. R. Dixon, I. Ya. Subbotin 37 The groups G in which the set Lnon−an(G) satisfies the minimal con- dition (namely the groups with the condition Min-(non-an)) were studied by L. A. Kurdachenko and V. V. Pylaev [40] and the following result was obtained. Theorem 3.2. (I) A non-periodic group G satisfies the condition Min- (non-an) if and only if it is either cental-by-finite or has a finite normal subgroup F such that H = G/F is a group of one of the following types: (1) H = A ⋊ ⟨b⟩, where ∣b∣ = p, for the prime p, A = CH(A) is a divisible abelian subgroup of 0-rank p − 1, and b induces a rationally irreducible automorphism on A; (2) H = K × L, where K is a divisible Chernikov group and L is a group of type (1). (II) A locally finite group G satisfies the condition Min-(non-an) if and only if it is either a Chernikov group or it is central-by-finite. Groups in which the set Lnon−an(G) satisfies the maximal condi- tion (namely the condition Max-(non-an)) were studied by L. A. Kur- dachenko, N. F. Kuzennyi and N. N. Semko in the article [37]. These groups are easier to describe. For example, a locally soluble group G satisfies the condition Max-(non-an) if and only if it is either polycyclic or central-by-finite. Groups in which the set Lnon−an(G) satisfies the weak minimal con- dition (respectively the weak maximal condition), namely the groups with the Min-∞-(non-an) condition, (respectively Max-∞-(non-an)) were stud- ied by G. Cutolo and L. A. Kurdachenko [15]. One of the main results of this paper is the following. Theorem 3.3. Let the group G have an ascending series of subgroups every factor of which is a locally (soluble-by-finite) group. If G satisfies Min-∞-(non-an) (respectively Max-∞-(non-an)) then either G/�(G) is finite or G is an almost soluble A3-group. To end this section, we note that S. Fransiosi, F. de Giovanni and L. A. Kurdachenko [27] studied groups G in which the set Lnon−an(G) consists of subnormal subgroups. 4. Groups with “small” systems of non-subnormal sub- groups It is well known that a finite group G with all subgroups subnormal (so Lnon−sn(G) is empty) is nilpotent. In infinite groups the situation is Jo u rn al A lg eb ra D is cr et e M at h .38 Finiteness conditions on some subgroup systems different. There are soluble locally nilpotent groups with trivial center all of whose subgroups are subnormal. Examples of such groups have been constructed by H. Heineken and I. Mohamed [31, 33], B. Hartley [29] and F. Menegazzo [62], among others. Groups with all subgroups subnormal have been studied in many papers and books; the book [60] by J. C. Lennox and S. E. Stonehewer is one excellent reference source. However since that book appeared there has been much activity in this area which we now describe. First, we must mention the following remarkable result due to W. Möhres [64]. Theorem 4.1. Let G be a group all of whose subgroups are subnormal. Then G is soluble. As we remarked above, an infinite group with all subgroups subnormal need not be nilpotent. However there are a number of cases in which nilpotence is assured in a group G when all subgroups are subnormal. We summarize some of what is known next. Theorem 4.2. Let G be a group with all subgroups subnormal. Then G is nilpotent if any one of the following hypotheses also hold. (1) G is periodic and hypercentral (W. Möhres [64]); (2) G is periodic and residually finite (H. Smith [82]); (3) G has a normal nilpotent subgroup A such that G/A is bounded (H. Smith [83]); (4) G is periodic and residually nilpotent (H. Smith [84], C. Casolo [5]); (5) G is torsion free (H. Smith [85], C. Casolo [4]). Some conditions for the nilpotency of a group with all subgroups subnormal are connected to properties of normal closures of elements and have been considered by L. A. Kurdachenko and H. Smith in [48]. Groups in which the set Lnon−sn(G) satisfies the minimal condition (the groups with the condition Min-(non-sn)) were considered by S. Fran- siosi and F. De Giovanni [25]. Under certain other hypotheses, these groups are either Chernikov or are groups with all subgroups subnormal. By contrast, the study of groups in which the set Lnon−an(G) satisfies the maximal condition (the groups with the condition Max-(non-sn)) turns out to be more interesting and such groups were considered by L. A. Kurdachenko and H. Smith in [45]. The main results of this paper can be summarized in the following theorem. Here we let B(G) denote the Baer radical of the group G, the subgroup of G generated by all cyclic subnormal subgroups of G. Jo u rn al A lg eb ra D is cr et e M at h .M. R. Dixon, I. Ya. Subbotin 39 Theorem 4.3. (i) A locally nilpotent group satisfies Max-(non-sn) if and only if all of its subgroups are subnormal; (ii) A locally soluble group G satisfies Max-(non-sn) if and only if it has one of the following types: (1) G is almost polycyclic; (2) each subgroup of G is subnormal; (3) G ∕= B(G) and G/B(G) is finitely generated, almost abelian and torsion free. In this case B(G) is nilpotent and for every element g /∈ B(G) every G-invariant abelian factor of B(G) is finitely generated as a ℤ ⟨g⟩-module. A class of groups that is closely related to the class Max-(non-sn) is the class of groups G in which the set Lnon−sn(G) consists of finitely generated subgroups. These groups have been considered by H. Heineken and L. A. Kurdachenko in [30]. The groups G for which Lnon−sn(G) satisfies the weak minimal con- dition (the groups with Min-∞-(non-sn)) were studied by L. A. Kur- dachenko and H. Smith in [46]. The main result of this work shows that the situation here is close to that arising for the condition Min-(non-sn). For example, it is shown in [46] that if G is a generalized radical group, that is, has an ascending series of subgroups every factor of which is either locally nilpotent or locally finite, then if G satisfies Min-∞-(non- sn), either all subgroups of G are subnormal or G is almost soluble and minimax. L. A. Kurdachenko and H. Smith [47] also discussed those groups G where the set Lnon−sn(G) satisfies the weak maximal condition (the groups with Max-∞-(non-sn)). In this case the situation is much more complicated. We finish this section by exhibiting a couple of results that illustrate this investigation. Theorem 4.4. (i) Let G be a locally finite group with Max-∞-(non- sn). Then either all subgroups of G are subnormal or G is a Chernikov group; (ii) Let G be a Baer group with Max-∞-(non-sn). Then all subgroups of G are subnormal. 5. Groups with “small” systems of non-abelian subgroups and other restrictions on non-abelian subgroups As we have already mentioned the description of the finite non-abelian groups all of whose proper subgroups are abelian (the groups G such that Jo u rn al A lg eb ra D is cr et e M at h .40 Finiteness conditions on some subgroup systems Lnon−ab(G) = {G}), due to G. Miller and H. Moreno [63], is one of the first important results of abstract group theory. Sophisticated examples of such infinite groups have been developed quite recently by A. Yu. Olshanskii (see the book [68, § 28]). These example suggest that it is impossible to expect a complete description of these groups. Groups in which the set Lnon−ab(G) satisfies the minimal condition (the groups with Min-(non-ab)) were first discussed by S. N. Chernikov [7]. His results imply that a non-abelian locally soluble group satisfying Min- (non-ab) is a Chernikov group, a result that has been extended to locally finite groups by V. P. Shunkov [80]. By contrast, the class of groups with Max-(non-ab) (which consists of those groups G in which Lnon−ab(G) satisfies the maximal condition) does not coincide with the class of groups satisfying Max, even when further stringent hypotheses are added. A simple example here is a group that is a wreath product of a group of prime order with an infinite cyclic group. Groups with Max-(non-ab) were considered somewhat later than groups with Min-(non-ab) by D. I. Zaitsev and L. A. Kurdachenko [91] and their main result is the following. Theorem 5.1. Let G be a locally (soluble-by-finite) non-polycyclic group. Then G satisfies the condition Max-(non-ab) if and only if it contains a normal abelian subgroup A with the following properties: (a) A = CG(A); (b) G/A is an infinitely generated, almost abelian, torsion-free group; (c) A is finitely generated ℤ⟨g⟩-module for all g ∈ G. The next step in the natural classification process here is the consider- ation of groups G in which the set Lnon−ab(G) satisfies the weak minimal condition (a class denoted by Min-∞-(non-ab)) and such groups were studied by D. I. Zaitsev in [89]. The main result of [89] shows that the situation here is similar to the case of the condition Min, since it is shown that a non-abelian almost soluble group G satisfies Min-∞-(non-ab) if and only if it is almost soluble minimax. Groups G in which the set Lnon−ab(G) satisfies the weak maximal condition (a class denoted by Max-∞-(non-ab)) were studied by L. S. Kazarin, L. A. Kurdachenko and I. Ya. Subbotin in [35]. The situation here is more complicated than in the cases of Max-(non-ab) and Min- ∞-(non-ab). However the non-abelian locally finite groups with Max- ∞-(non-ab) are minimax (and hence Chernikov groups). The description of other classes of groups with Max − ∞ − (non − ab) requires special definitions and terminology and we now discuss these here. Jo u rn al A lg eb ra D is cr et e M at h .M. R. Dixon, I. Ya. Subbotin 41 Let G be a group and let A be a normal abelian subgroup of G. Let aG(A) denote the G–invariant subgroup of A satisfying the conditions: (i) aG(A) has an ascending series of G–invariant subgroups whose fac- tors are G–chief; (ii) A/aG(A) has no nontrivial minimal G–invariant subgroups. Let R be a ring and let A be an R–module. We say that A is R– minimax if A has a finite series of submodules, every factor of which is either Artinian or Noetherian. Generalized radical groups were studied in [35] and the following theorems represent the main results of that work. Theorem 5.2. Let G be a non–abelian generalized radical group, let A be a maximal normal abelian subgroup of G and let T be the periodic part of A. Suppose that either aG(A) does not contain T or that r0(A) is infinite. Then G satisfies Max-∞-(non-ab) if and only if the following conditions hold: (i) G/A is torsion–free, finitely generated and abelian–by–finite; (ii) For each element g ∈ G∖A the ℤ ⟨g⟩–module A is minimax. Theorem 5.3. Let G be a non–abelian generalized radical group, let A be a maximal normal abelian subgroup of G and let T be the periodic part of A. Suppose that A is non– minimax, T ≤ aG(A) and r0(A) is finite. Then G satisfies Max-∞-(non-ab) if and only if the following conditions hold: (i) A/T is minimax; (ii) G/A is torsion–free; (iii) G/A = L contains a normal subgroup K = H/A of finite index such that either K is abelian and minimax or K = C ⋋ D where C = CK(C) is abelian and minimax, D = CK(D) is abelian and finitely generated; (iv) For each element g ∈ G∖A the ℤ ⟨g⟩–module T is Artinian. When the maximal normal abelian subgroup A is minimax these re- sults take a particularly pleasing form, as follows. Theorem 5.4. Let G be a non–abelian generalized radical group and let A be a maximal normal abelian subgroup of G. Suppose that A is minimax. Then G satisfies Max-∞-(non-ab) if and only if G is soluble– by–finite and minimax. Jo u rn al A lg eb ra D is cr et e M at h .42 Finiteness conditions on some subgroup systems Those groups G having finite derived subgroup (the so-called BFC– groups) are a natural generalization of abelian groups. We let Lnon−BFC(G) denote the family of groups all of whose subgroups have infinite derived subgroups. In the papers [19, 20] of M. R. Dixon and L. A. Kurdachenko, the groups in which the family Lnon−BFC(G) satisfies the maximal condi- tion (the groups satisfying Max-(non-BFC)) were studied. The following theorems describe the main results of [19]. If G is a group and F is the class of finite groups then we let GF denote the finite residual of G, the intersection of all subgroups of G of finite index in G. Theorem 5.5. (1) Let G be a locally finite group satisfying Max- (non-BFC). If G/GF is finite and [G,G] is infinite, then G is a Chernikov group. (2) Let G be a locally FC–group satisfying Max-(non-BFC). If G/GF is not finitely generated and [G,G] is infinite, then G has a series of normal subgroups F ≤ T ≤ L ≤ G such that (i) F is finite; (ii) T = FD where D = GF ≤ �(G) is a divisible Chernikov p– subgroup for some prime p; (iii) G/T is torsion–free abelian and L/T is finitely generated; (iv) L/F is abelian, G/L is a Prüfer p–group and T/F = [G/F,G/F ]; (v) The p–rank of D is at most the 0–rank of L/T ; (vi) If H is a subgroup of G having infinite derived subgroup, then D ≤ H. Theorem 5.6. Let G be a non–periodic nilpotent locally FC–group satisfying Max-(non-BFC). If G/GF is finitely generated and [G,G] is infinite, then (i) G/GF is a BFC–group; (ii) GF = P ×Q where P,Q are Prüfer p–groups for some prime p; (iii) G/CG(G F) is a torsion–free abelian group and [GF, g] = P ≤ �(G) for each g ∈ C∖CG(G F). Theorem 5.7. Let G be a non–nilpotent locally FC–group satisfying Max-(non-BFC). If G/GF is finitely generated and [G,G] is infinite then either G contains a finite normal subgroup F such that G/F is nilpotent, or G has a series of normal subgroups D ≤ C ≤ G such that (i) D = GF is a divisible Chernikov p–subgroup for some prime p; Jo u rn al A lg eb ra D is cr et e M at h .M. R. Dixon, I. Ya. Subbotin 43 (ii) G/D is a BFC−group; (iii) C = CG(D) and if g ∈ G∖C then every proper ⟨g⟩–invariant sub- group of D is finite; (iv) If P/C is the periodic part of G/C then either P/C has order p or P/C is a cyclic p′–group. In the former case, D has rank p−1, and in the latter case D has rank at most q− 1 for all primes q dividing the order of P/C. Furthermore L/F is abelian, G/L is a Prüfer p–group and T/F = [G/F,G/F ]. (v) The p–rank of D is at most the 0–rank of L/T . (vi) If H is a subgroup of G having infinite derived subgroup, then D ≤ H. The following two theorems constitute the main results of [20]. Theorem 5.8. Let G be a finitely generated soluble–by–finite group satisfying Max-(non-BFC). Suppose that S is the soluble radical of G and that K is that term of the derived series of S such that S/K is finitely generated but K/[K,K] is not finitely generated. Suppose also that K is a BFC–group. Then G has a series of normal subgroups F ≤ A ≤ G such that (i) F is finite; (ii) A/F is abelian; (iii) G/A is finitely generated, abelian–by–finite and torsion–free; (iv) A/F is a finitely generated ℤ ⟨g⟩–module for each element g ∈ G∖A. Theorem 5.9. Let G be a finitely generated soluble–by–finite group satisfying Max-(non-BFC). Suppose that S is the soluble radical of G and that K is that term of the derived series of S such that S/K is finitely generated but K/[K,K] is not finitely generated. Suppose also that [K,K] is infinite. Then G has a series of normal subgroups F ≤ T ≤ A ≤ G such that (i) F is finite; (ii) T = FD where D ≤ �(A) is a divisible Chernikov p–subgroup for some prime p; (iii) A/T is abelian and torsion–free; Jo u rn al A lg eb ra D is cr et e M at h .44 Finiteness conditions on some subgroup systems (iv) G/A is finitely generated, abelian–by–finite and torsion–free; (v) A/T is a finitely generated ℤ ⟨g⟩–module for each element g ∈ G∖A; (vi) if the subgroup H has infinite derived subgroup, then H contains D. In the paper [21] a more general situation has been considered, that in which the family Lnon−FC(G), of all non–FC–subgroups, satisfies the maximal condition. This is the class of groups with Max-(non-FC). If G is a locally FC–group then the set of elements of finite order forms a subgroup, the torsion subgroup of G, which we denote by Tor(G). The derived subgroup of an FC–group is well–known to be periodic, so if G is a locally FC–group then G/Tor(G) is torsion-free abelian. The following result holds. Theorem 5.10. (i) Let G be a locally FC–group satisfying Max- (non-BFC). If G/Tor(G) is not finitely generated then either G is an FC–group or G satisfies Max-(non-BFC); (ii) Let G be a locally FC–group satisfying Max-(non-BFC). If G is soluble, then either G is an FC–group or G satisfies Max-(non- BFC). 6. Groups with “small” systems of non-nilpotent subgroups and related topics As we have already mentioned, the description of the finite non-nilpotent groups in which all proper subgroups are nilpotent (the groups with Lnon−nil(G) = {G}) was obtained by O.Yu. Schmidt [73]. We next con- sider some generalizations of this work concerned with infinite groups. First, we discuss the work of M. F. Newman and J. Wiegold [67]), who studied the class of groups G in which, for some fixed natural number k, Lnon−nil(k)(G) = {G}. They showed that then G can be generated by at most k + 1 elements. Furthermore, they proved that if Lnon−nil(k)(G) = {G} or Lnon−nilG = {G} then G/Fratt(G) is a non-abelian simple group. Also, for a simple group G in which Lnon−nil(k)(G) = {G} or Lnon−nil(G) = {G}, it is the case that (a) every pair of maximal subgroups of G has trivial intersection; (b) if 1 ∕= x ∈ G, then there is an element g, such 〈 g−1xg, x 〉 = G; (c) G has no involutions. Jo u rn al A lg eb ra D is cr et e M at h .M. R. Dixon, I. Ya. Subbotin 45 Examples of such simple groups have been obtained by A.Yu. Ol- shanskii [68, Section 28]. Following D. I. Zaitcev [86], we call a nilpotent group G, of nilpotency class k, stably nilpotent if every infinite subgroup of G (in particular G itself) of nilpotency class k contains a proper subgroup of nilpotency class k. It turns out that every infinite nilpotent group of nilpotency class k contains a proper infinite subgroup of nilpotency class k and that if G is a nilpotent torsion–free group, then every nontrivial subgroup is stably nilpotent. This means that if G is a nilpotent group in which Lnil(k)(G) = {G} then G is finite. These results are due to D. I. Zaitsev [86]. D. I. Zaitsev continued this investigation in [87] where he showed that a locally nilpotent group G which has a nilpotent subgroup of nilpotency class k contains a stably nilpotent subgroup of class k if and only if G is not a Chernikov group (D.I. Zaitcev [87]). A. N. Ostylovskii took up a similar theme in [69] where locally finite groups (and more generally binary finite groups) were considered. He proved in this case that if every infinite subgroup of G that has infinite index either satisfies Min or has nilpotency class at most k then either G satisfies Min or is nilpotent of class at most k. Furthermore, if G is a binary finite group that is not of nilpotency class at most k and if the set Lnil(k)(G) satisfies the weak minimal condition, then G is a Chernikov group. The examples constructed by A. Yu. Olshanskii leave little hope of obtaining a complete description of finitely generated such groups. However, a recent amazing theorem of Asar [1] shows that every locally graded group, all of whose proper subgroups are nilpotent, is soluble. The following result supplements this theorem of Asar very nicely and was obtained by H. Smith in [81]. Theorem 6.1. Let G be a soluble non-nilpotent group all of whose proper subgroups are nilpotent. If G has no maximal subgroups, then the following conditions hold: (a) G is a countable p-group for some prime p; (b) G/[G,G] is a Prufer p-group; (c) every subgroup of G is subnormal; (d) [G,G]p ∕= [G,G] and every hypercentral subgroup G is abelian; in particular, [G,G] = n(G) for all n ≥ 2; (e) �(G) contains all divisible subgroups of G; Jo u rn al A lg eb ra D is cr et e M at h .46 Finiteness conditions on some subgroup systems (f) CG([G,G]) is abelian and [G,G] is a non-essential subgroup (thus if H[G,G] = G then H = G for every H ≤ G); in particular G contains no proper subgroups of finite index; (g) if H is finite subgroup of [G,G], then HG ∕= [G,G]; (h) the hypercenter of G coincides with its center. H. Smith [81] also initiated the investigation of those groups G in which the set Lnon−nil(G) satisfies the maximal, minimal, weak minimal and weak maximal conditions. For example a locally nilpotent torsion- free group satisfying one of these conditions is nilpotent. The groups G for which Lnon−nil(G) satisfies the maximal condition have also been studied by M. R. Dixon and L. A. Kurdachenko in [17, 18]. The first article deals with locally nilpotent such groups, while the second article is dedicated to soluble groups. Here are the main results of the first article. Theorem 6.2. Let G be a locally nilpotent group satisfying the condition Max-(non-nil) and let T be its periodic part. If G is non-nilpotent and G/GF is infinitely generated, then the following conditions hold: (a) GF ≤ T and T/GF is finite; (b) G/T is a nilpotent maximal subgroup and Sp(G/T ) = {p} for some prime p; (c) GF is a p-subgroup; (d) G contains a nilpotent normal subgroup U such that G/U is a Prüfer p-group; (e) if S is a non-nilpotent subgroup of G, then G = US. Theorem 6.3. Let G be a locally nilpotent group satisfying the condition Max-(non-nil). If G is non-nilpotent, G/GF is finitely generated and GF is nilpotent, then the following conditions hold: (a) GF is a divisible Chernikov group; (b) every proper G-invariant subgroup of G is finite; (c) [G,GF] = GF. Theorem 6.4. Let G be a locally nilpotent group satisfying the condi- tion Max-(non-nil) and let T be its periodic part. If G is non-nilpotent and non-minimax and G/GF is infinitely generated, then the following conditions hold: Jo u rn al A lg eb ra D is cr et e M at h .M. R. Dixon, I. Ya. Subbotin 47 (a) GF is periodic; (b) GF has no proper subgroup of finite index; (c) GF is non-nilpotent while all its proper subgroups are nilpotent; (d) G is soluble; (e) GF is nilpotent-by-Chernikov. In particular, GF is a p-subgroup for some prime p, and it has an ascending series of nilpotent G-invariant subgroups ⟨1⟩ = A0 ≤ A1 ≤ ⋅ ⋅ ⋅ ≤ An ≤ ⋅ ⋅ ⋅ ≤ ∪ n∈ℕ An = GF. The main results of the article [18] are as follows. Theorem 6.5. Let G be a locally (soluble–by–finite) group satisfying Max-(non-nil) and let L be the locally nilpotent radical of G. If L ∕= G, then L is nilpotent and G/L is a finitely generated abelian–by–finite group. Theorem 6.6. Let G be a locally (soluble–by–finite) group satisfying Max-(non-nil) and let L be the locally nilpotent radical of G. Suppose that L is not finitely generated. Let g be an element of G∖L such that gL has infinite order. Let ⟨1⟩ = L0 ≤ L1 ≤ ⋅ ⋅ ⋅ ≤ Ln = L be the upper central series of L and for k > n write Lk = Ln. Then (i) ⟨L, g⟩ is non–nilpotent and finitely generated; (ii) there exists m < n such that ⟨Lm, g⟩ is locally nilpotent but ⟨Lm+1, g⟩ is not locally nilpotent; (iii) every factor Lj+1/Lj is finitely generated as a ℤ ⟨g⟩–module for j ≥ m; (iv) either ⟨Lm, g⟩ is nilpotent or Lm contains a central divisible Chernikov subgroup D such that Lm/D is finitely generated and D = [D, g]. The study of those groups G in which the set Lnon−nil(G) satisfies the weak maximal condition was initiated by L. A. Kurdachenko, P. Shumy- atskii and I. Ya. Subbotin in [44]. The main results of this paper are as follows. Jo u rn al A lg eb ra D is cr et e M at h .48 Finiteness conditions on some subgroup systems Theorem 6.7. Let G be a locally finite group satisfying Max-∞-(non- nil). If G is not locally nilpotent, then G is Chernikov. Theorem 6.8. Let G be a locally nilpotent group satisfying Max-∞- (non-nil). Suppose that G is non–nilpotent. Then G satisfies the follow- ing conditions: (i) G/GF is nilpotent and minimax; (ii) the set Π(G) is finite; (iii) the subgroup GF is periodic; (iv) if the subgroup M = [GF, GF] is non–nilpotent, then GF/M is a divisible Chernikov group and every proper G–invariant subgroup of M is nilpotent. In particular, M is a p–subgroup for some prime p, and M has an ascending series of G–invariant subgroups ⟨1⟩ = A0 ≤ A1 ≤ ⋅ ⋅ ⋅ ≤ An ≤ ⋅ ⋅ ⋅ ≤ ∪ n∈ℕ An = M such that every subgroup An is nilpotent. Theorem 6.9. Let G be a generalized radical group satisfying Max- ∞-(non-nil) and let L be the locally nilpotent radical of G. Then L is nontrivial. If L is nilpotent, then G/L is minimax, soluble–by–finite and almost torsion–free. Theorem 6.10. Let G be a generalized radical group satisfying Max- ∞-(non-nil) and let L be the locally nilpotent radical of G. If L is non– nilpotent, then G satisfies the following conditions: (i) G/GF is minimax, soluble–by–finite and almost torsion–free; (ii) the set Π(G) is finite; (iii) GF = LF is periodic; (iv) if the subgroup M = [GF, GF] is non–nilpotent, then GF/M is a divisible Chernikov group and every proper G–invariant subgroup of M is nilpotent. The study of groups satisfying Max-∞-(non-nil) was continued in the article [43] of L. A. Kurdachenko and N. N. Semko, where they described the hypercentral groups of this kind. Here is the main result of that work: Jo u rn al A lg eb ra D is cr et e M at h .M. R. Dixon, I. Ya. Subbotin 49 Theorem 6.11. Let G be a hypercentral group satisfying Max-∞-(non- nil). Suppose that G is non–nilpotent and non–minimax. Then G con- tains a finite normal subgroup F such that G/F ≤ M ×L where M is a hypercentral minimax group and L satisfies the following conditions: (i) L is a hypercentral non–nilpotent, non–minimax group satisfying Max−∞− (non− nil); (ii) the periodic part P of the group L has a central series of L– invari- ant subgroups ⟨1⟩ = L0 ≤ L1 ≤ ⋅ ⋅ ⋅ ≤ Ln = P such that the factors Lj+1/Lj , for 0 ≤ j ≤ n − 2, are elementary abelian and P–quasifinite, Ln/Ln−1 is finite, and, in particular, P is a bounded nilpotent subgroup; (iii) L/P is abelian–by–finite and minimax. 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E-Mail: isubboti@nu.edu Received by the editors: 06.07.2009 and in final form 06.07.2009.

Groups with finiteness conditions on some subgroup systems: a contemporary stage

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