# On metanilpotent groups satisfying the minimal condition on normal subgroups

Derek J. S. Robinson
From the journal Journal of Group Theory

## Abstract

A comprehensive account is given of the theory of metanilpotent groups with the minimal condition on normal subgroups. After reviewing classical material, many new results are established relating to the Fitting subgroup, the Hirsch–Plotkin radical, the Frattini subgroup, splitting and conjugacy, the Schur multiplier, Sylow structure and the maximal subgroups. Module theoretic and homological methods are used throughout.

## 1 Introduction

The minimal condition on subgroups, min, was introduced into group theory during the 1940s. Among the early investigators was S. N. Černikov, who proved that a soluble group satisfying min is an extension of a divisible abelian group of finite rank by a finite group [6]; groups with this structure are nowadays called Černikov groups. Subsequently the property min was studied for many other classes of infinite groups.

The minimal condition for normal subgroups min-n, which is clearly a much weaker property, was first studied by R. Baer in a paper [3] that appeared in 1949. In the same year V. S. Čarin [5] gave the first example of a metabelian group with min-n that is not a Černikov group. Then, in 1964, Baer [4] established the fundamental fact that soluble groups with min-n are locally finite. He also observed that a nilpotent group with min-n satisfies min and indeed is a Černikov group with centre of finite index – see [16, Theorem 3.14]. However, since that time, there has been relatively little progress in research on soluble groups with min-n in general.

On the other hand, the study of metabelian groups with min-n has proved to be more amenable. In 1970, D. McDougall [14] was able to show that these groups are countable. Subsequent work of McDougall, B. Hartley and H. L. Silcock in the 1970s [8, 9, 19, 20, 21] established the basis of the theory. It was soon observed that many results obtained for metabelian groups are in fact valid for metanilpotent groups, i.e., soluble groups with nilpotent length at most 2. In investigations of this class of groups, it became apparent that artinian modules over nilpotent Černikov groups must play a prominent role. In this respect, the fundamental paper [9] of Hartley and McDougall has proved invaluable.

After the burst of activity in the 1970s, little has been done on metanilpotent groups with min-n. Nevertheless, the topic remains an attractive area for investigation, calling for the use of module theoretic and homological methods, as well as established techniques from infinite soluble group theory.

Our object in this paper is twofold. First we describe the classical theory of metanilpotent groups with min-n, providing improved proofs where possible. Then we establish many new results. In particular, we consider the structure of the Fitting subgroup and the Hirsch–Plotkin radical, showing that these are respectively nilpotent and hypercentral. It is shown that the Schur multiplier of a metanilpotent group with min-n is always finite, and in many cases is zero. It turns out that metanilpotent groups with min-n have a surprisingly satisfactory Sylow theory; indeed, in every subgroup, the Sylow π-subgroups are conjugate. Necessary and sufficient conditions are found for the Frattini subgroup to be nilpotent or locally nilpotent. It is proved that the limit of the lower central series has a nilpotent Černikov supplement with finite intersection. For groups that have no proper subgroups of finite index, there is even a splitting and conjugacy theorem. The maximal subgroups of metanilpotent groups with min-n are studied, with particular attention to their cardinality. Finally, we describe recent joint work with A. Arikan and G. Cutolo [2] on countable domination of the proper subgroups of metanilpotent groups with min-n.

In much of the theory, a direct decomposition of artinian modules over locally finite groups due to Hartley and McDougall plays a prominent role in non-modular situations. In addition, we make use of a near direct decomposition established in [2], which is valid in the modular case.

It is hoped that this article will help to stimulate research in a very fertile area of infinite soluble group theory.

### Notation

1. Gab: the abelianization

2. G(i): a term of the derived series

3. A[n]: the subgroup of elements with orders dividing n in an abelian group

4. π(G): the set of primes dividing orders of elements of G

5. Sylπ(G): the set of Sylow π-subgroups of G

6. γi(G), γ(G): terms of the lower central series

7. Zi(G): a term of the upper central series

8. Z(G) and Z¯(G): the centre and hypercentre

9. Fitt(G): the Fitting subgroup

11. A1A2An: the near direct sum of submodules Ai

12. Der(G,A), Inn(G,A): groups of derivations and inner derivations

13. AQ, AQ: the set of Q-fixed points and the Q-trivialization of a Q-module A

Modules are understood to be right modules.

### Index of topics

1. Basic results

2. Module theoretic methods

3. Structural applications

4. Groups without proper subgroups of finite index

5. The Schur multiplier

6. Sylow properties

7. Frattini subgroups

8. Maximal subgroups

9. Countable domination

## 2 Basic results

We begin with a fundamental theorem of Baer, which is still the only general result known for soluble groups with min-n.

## Theorem 2.1 (Baer [4]).

A soluble group with min-n is locally finite.

The proof depends on the simple result about automorphism groups that follows; for a short proof, see [12, 1.5.3].

## Lemma 2.2 (Baer [4]).

Let G be a locally finite group of automorphisms of an abelian group A. If A is simple as a G-module, then it is an elementary abelian p-group for some prime p.

## Proof of Theorem 2.1.

Let G be a soluble group with min-n; we can assume the derived length d exceeds 1. Put A=G(d-1), and note that Q=G/A is locally finite by induction on d. Since G has min-n, there is an ascending Q-composition series in A. If F is a factor of this series, it is a simple Q-module, and in addition, Q/CQ(F) is locally finite. It follows from Lemma 2.2 that F is elementary abelian, and hence A is periodic. Therefore, G is locally finite. ∎

We remark that H. Heineken and J. S. Wilson [10] have constructed a non-trivial locally soluble group with min-n that is torsion-free. From now on, we will deal almost exclusively with metanilpotent groups. The next result is fundamental.

## Theorem 2.3.

A metanilpotent group with min-n is countable.

This was proved in the metabelian case by McDougall [14], while the general result by Silcock appears in [19, Theorem C].

## Proof of Theorem 2.3.

Let G be a metanilpotent group with min-n, and write the final term of the lower central series as N=γ(G); thus N and G/N are nilpotent by min-n. Also, G/N is a Černikov group, so it is countable and has a divisible abelian normal subgroup of finite index, say F/N. By a result of Wilson [23], the subgroup F also satisfies min-n. If Nab is countable, then, by the well-known tensor product property of the lower central series (see [15]), the subgroup N, and hence G, is countable. Thus, factoring out by N and replacing G by F, we can assume that G is metabelian.

Set A=G, and observe that A is an artinian module over Q=G/A by conjugation. Let aA, and put I=AnnQ(a), the annihilator of a in Q; then (a)Q𝑄Q/I and Q/I is a commutative artinian ring. By a well-known theorem of Hopkins, Q/I is noetherian, and therefore (a)Q has finite -composition length. From this, it follows that the length of the upper socle series of the module A is at most ω. Since Q is countable, simple Q-modules are countable. Thus A, and hence G, is countable. ∎

On the other hand, Hartley [8] has constructed uncountable soluble groups of derived length 3 which have min-n. This is the first indication that things can go wrong when we try to extend results proven for metanilpotent groups to soluble groups with derived length greater than 2.

The final result in this section is surely widely known, but it does not seem to appear explicitly in the literature.

## Lemma 2.4.

Let G be a periodic nilpotent group, and let D be a subgroup of G which has no proper subgroups of finite index. Then D is divisible and DZ(G).

## Proof.

Let c be the nilpotent class of G. If c1, then clearly D is divisible. Let c>1, and proceed by induction on c. Then DZ(G)/Z(G) is divisible and is contained in Z2(G)/Z(G). Let gG and dD. If g has order n, we can write d=d1nz, where d1D and zZ(G). Since [g,d1]Z(G), we have 1=[gn,d1]=[g,d1]n=[g,d1n]=[g,d]. Therefore, DZ(G) and of course D is divisible. ∎

## 3 Module theoretic methods

Let G be a metanilpotent group with min-n, and let NG with N and Q=G/N nilpotent. Then Nab is an artinian module over the nilpotent Černikov group Q. Since this module has a major influence on the group, it is essential to investigate the structure of artinian modules over nilpotent Černikov groups. Some initial information is given by the following simple result.

## Lemma 3.1.

Let A be an artinian module over a group Q. Then the following hold.

1. There is an expression A=D+B with D and B submodules, where D is divisible and B is bounded as abelian groups. Moreover, π(A) is finite.

2. If Q is a nilpotent Černikov group, then A is a countable periodic abelian group.

## Proof.

(i) Since A is artinian, there is an integer n>0 such that

n!A=(n+1)!A==D,

say. Clearly, D is a divisible abelian group. If aA, we have n!a=n!d for some dD. Thus n!(a-d)=0 and a-dA[n!], showing that A=D+A[n!]. Also, π(A) is finite by the artinian property.

(ii) Form the semidirect product G=QA, observing that G is a metanilpotent group with min-n. The statement follows at once from Theorem 2.3 and Theorem 2.1. ∎

### Module constructions

We will describe a method for constructing uniserial artinian modules over certain locally finite groups, starting from simple modules; here we follow ideas of Hartley and McDougall [9]. Let p be a prime and Q a countable centre-by-finite p-group. Denote by {MλλΛ} a complete set of non-isomorphic simple pQ-modules; thus we are in a non-modular situation. Here Mλ is an elementary abelian p-group of rank rλ, say. Choose a divisible abelian p-group Vλ of rank rλ, and identify Mλ with Vλ[p], thereby endowing the latter with a Q-module structure. Since Q is a countable locally finite p-group, the module structure of Vλ[p] may be extended to Vλ – see for example [9, Lemma 3.2]. The resulting pQ-module will be denoted by

Vλ().

It is not hard to show that the only proper submodules of Vλ() are the

Vλ(n)=Vλ[pn],

where n=0,1,2,. Thus Vλ() is a uniserial artinian Q-module, while Vλ(n) is noetherian and artinian. In addition,

Vλ(n+1)/Vλ(n)𝑄Mλ.

Keep in mind that Vλ() is a divisible abelian p-group; in fact, it is the injective hull of Mλ, which is unique up to isomorphism. For these facts, see [9, § 2, 3].

Notice that the module Vλ() is p-adically irreducible, i.e., it is a p-group that is unbounded, but every proper submodule is bounded. These observations permit the construction of uniserial artinian modules over nilpotent Černikov groups in the non-modular case since such groups are centre-by-finite.

### Modules over abelian groups

In the case of abelian groups, it is easy to describe the simple modules explicitly. The following approach is well known. Let p be a prime and Q an abelian p-group. Let θ:Q(¯p) be a homomorphism, where ¯p is the algebraic closure of the field of p elements and the asterisk denotes the multiplicative group. Denote by Fθ the additive subgroup of ¯p which is generated by Im(θ). Then Fθ is a subfield of ¯p, which becomes a Q-module via the action fx=fxθ, where fFθ, xQ. It is easily verified that Fθ is a simple pQ-module, and it is well known that every simple pQ-module is isomorphic with some Fθ. In addition, Fθ𝑄Fϕ if and only if Im(θ)=Im(ϕ) and θ=ϕρ, where ρGal(Fθ/p) – see [9, Lemma 2.5].

Next suppose that, in addition, Q is a Černikov group. Then Gθ=QFθ is a metabelian group with min-n, and Gθ will be non-Černikov if and only if Im(θ) is infinite.

For example, let π be a finite set of primes not containing the prime p. Then ¯p contains primitive qi-th roots of unity, where i=1,2, and qπ. Let Q be the subgroup of (¯p) generated by these roots; thus Q is a locally cyclic, divisible abelian π-group. Let θ:Q(¯p) be inclusion. Then Gθ is a metabelian group with min-n. We will call Fθ a Čarin (p,π)-module and Gθ a Čarin (p,π)-group; in fact, when π consists of a single prime, Gθ is Čarin’s original example.

The construction of uniserial artinian Q-modules given above can be applied to a simple Q-module Fθ, if Q is an abelian p-group, to produce the modules

Vθ(n),Vθ(),n=0,1,2,.

Of course, Vθ(n+1)/Vθ(n)𝑄Fθ.

### The Hartley–McDougall decomposition

The significance of the modules Vλ(n) and Vλ() is made clear by the following crucial result of Hartley and McDougall [9, Theorem A]. We retain the notation of the preceding discussion.

### Theorem 3.2.

Let p be a prime and Q a countable centre-by-finite p-group. Let A be an artinian Q-module which is a p-group. Then A is the direct sum of finitely many modules of types Vλ(n),Vλ(). Moreover, the direct decomposition is unique up to an automorphism of A.

The theorem provides information on the structure of an artinian module A over a nilpotent Černikov group Q provided that π(A)π(Q) is empty, i.e., we are in the non-modular case. While the Hartley–McDougall decomposition cannot be applied in modular situations, this loss is diminished by the next result.

### Lemma 3.3.

If A is an artinian module over a nilpotent Černikov group Q, then Qp/CQp(Ap) is finite for all pπ(Q).

### Proof.

We can assume that A is a p-group and Q acts faithfully on A, so what needs to be proved is that Qp is finite. Assume this is false, so that the maximum divisible subgroup D of Qp satisfies 1<DZ(Q). Suppose first of all that A is a simple Q-module, and let dD have order p. Then

0=A(dp-1)=A(d-1)p

since A is elementary abelian. From DZ(Q), we deduce that each A(d-1)i is a Q-submodule, and thus A(d-1)=0 by the simplicity of A, a contradiction.

Reverting to the general case, we form an ascending composition series

{Aαα<β}in theQ-moduleA,

which is possible because A is artinian. There is a least α such that D acts non-trivially on Aα and α cannot be a limit ordinal. Hence D acts trivially on Aα-1, as well as on the simple module Aα/Aα-1 by the previous paragraph. Therefore, there is an injective homomorphism D/CD(Aα)Hom(Aα/Aα-1,Aα-1). But the latter group is elementary abelian, while D is divisible, so it follows that D=CD(Aα), a contradiction. ∎

Here is a first application of these methods.

### Proposition 3.4.

Let A be an artinian module over a nilpotent Černikov group Q. If A is bounded as an abelian group, then it is noetherian.

### Proof.

We can assume that A is a p-group and Q acts faithfully on it. Lemma 3.3 shows that Q=P×R, where P is a finite p-group and R is a p-group. Now QA satisfies min-n, and in addition |QA:RA|=|P| is finite, so by Wilson’s theorem RA also has min-n and A is R-artinian. Thus Theorem 3.2 is applicable and, as A is bounded, it is a direct sum of finitely many R-modules of the form Vλ(n). Each of the latter is R-noetherian, whence it follows that A is R-noetherian and hence Q-noetherian. ∎

This has immediate and significant consequences.

### Lemma 3.5.

Let G be a metanilpotent group with min-n. Let NG, where N and G/N are nilpotent. Then N and N/Z(N) have finite exponent and satisfy max-G.

### Proof.

By Lemma 3.1, the abelian group Nab is the direct product of a divisible subgroup and a subgroup of finite exponent. The tensor product property of the lower central series shows that γi(N)/γi+1(N) has finite exponent for i2, and hence it satisfies max-G by Proposition 3.4. Since N is nilpotent, N has finite exponent and max-G.

Next let D=n=1,2Nn!. Since G has min-n, N/D has finite exponent, and D has no proper subgroups of finite index. It follows from Lemma 2.4 that D is divisible, and DZ(N). Hence N/Z(N) has finite exponent and satisfies max-G. ∎

We will also need a decomposition that is valid in the modular case, as a replacement for the direct decomposition of Hartley and McDougall. As is usual in modular situations, some level of imprecision is inevitable; specifically, direct sums are replaced by near direct sums. Here a module A is said to be the near direct sum of submodules Ai, i=1,2,,n, if A=i=1nAi and each intersection Aij=1,jinAj is bounded as an abelian group. The notation

A=A1A2An

will be used to denote a near direct sum. The following result provides a near direct decomposition that is valid for any artinian module over a nilpotent Černikov group. It appears in a recent work of Arikan, Cutolo and Robinson [2, Proposition 5.5].

### Theorem 3.6.

Let p be a prime and A an artinian Q-module that is a p-group, where Q is a nilpotent Černikov group. Then there is a near direct decomposition

A=(A1A2An)+A[p]

in which the Ai are p-adically irreducible Q-modules and 0.

The theorem is proved by a variation on the averaging argument in the standard proof of Maschke’s theorem. We record a straightforward consequence.

### Corollary 3.7.

With the notation of Theorem 3.6,

A/A[pk]𝑄A1A2An

for some k0. If A is a divisible group, then A𝑄A1A2An.

### Remark.

In Theorem 3.6, the p-adically irreducible Q-submodules Ai are artinian as Qp-modules since each Qp/CQp(Ai) is finite by Lemma 3.3. Therefore, Ai is a direct sum of finitely many isomorphic uniserial injective Qp-modules of some type Vλ() by Theorem 3.2.

## 4 Structural applications

We will now apply the methods developed in Section 3 to investigate the structure of metanilpotent groups with min-n.

## Theorem 4.1.

Let G be a soluble group with min-n. Then the following hold.

1. The Fitting subgroup of G is a hypercentral group.

2. If G is metanilpotent, then the Baer radical of G is nilpotent and coincides with the Fitting subgroup.

## Proof.

Let F=Fitt(G), the Fitting subgroup of G. Let d>1 be the derived length of G, and set A=G(d-1). Then F/A is hypercentral by induction on d. Next use min-n to form an ascending G-composition series in A, say {Aαα<β}. If N is a nilpotent normal subgroup of G, then NA1 is nilpotent. Thus we have [A1,N]<A1 and hence [A1,N]=1, which shows that [A1,F]=1. By this argument, F centralizes every factor of the series, so that AZ¯(F). Therefore, F is a hypercentral group.

Now assume that G is metanilpotent. Let R denote the Baer radical of G, and put A=γ(G); thus A is nilpotent, so AR. If R/A is nilpotent, a well-known theorem of P. Hall shows that R is nilpotent. Thus we may assume that A is abelian. Furthermore, if R/Ap is nilpotent for all pπ(A), then R is nilpotent since π(A) is finite. Hence we can also assume that A is a p-group.

Next R is locally nilpotent, so it has a primary decomposition and clearly Rp centralizes A. Thus it remains only to prove that Rp acts nilpotently on A since R/A is nilpotent. Now A is G-artinian, so by Lemma 3.1 it has a divisible submodule D such that A/D has finite exponent. Let xR; then x is subnormal in G, and therefore

[D,x,,xn]=1for somen>0.

Since x has finite order and D is divisible, it follows that [D,x]=1, which shows that [D,R]=1. Thus we may assume that D=1 and hence that A has finite exponent. From Lemma 3.3, we know that Rp/CRp(A) is a finite p-group. A straightforward argument shows that Rp acts nilpotently on A. ∎

We can now give a detailed description of the structure of the Fitting subgroup.

## Theorem 4.2.

Let F denote the Fitting subgroup of a metanilpotent group G with min-n. Then the following hold.

1. There exists SG containing F such that S has finite exponent and satisfies max-G, while F/S is the direct sum of finitely many p-adically irreducible G/F-modules for various primes p.

2. If D denotes the maximum divisible subgroup of F, then DZ(F) and D is the direct sum of finitely many p-adically irreducible G/F-modules for various primes p, while F/D has finite exponent and satisfies max-G.

This follows on applying Lemmas 3.5 and 2.4, and Corollary 3.7 to the subgroup F. Next we turn to the Hirsch–Plotkin radical.

## Theorem 4.3.

Let G be a metanilpotent group with min-n. The Hirsch–Plotkin radical of G is then a hypercentral group, so it coincides with the hypercentral radical.

## Proof.

Let H=HP(G), the Hirsch–Plotkin radical of G, and write A=γ(G); thus AH. If H/A is locally nilpotent, then so is H by [15, Theorem 3]. Therefore, we can assume A to be abelian. Since G/A is countable and centre-by-finite, there is an ascending series A=H0H1 such that HiG, Hi/A is finite and H=i=1,2,Hi. Let 1aA; then aHi is finite and Hi is locally nilpotent, so aHiZ(Hi)1 and hence 1<Ci=CA(Hi)G. Since C1C2 and A satisfies min-G, there is an i0 such that Ci=Ci+1=. Hence Ci=CA(H)1. By factoring out by CA(H) and repeating the argument, we generate an ascending G-invariant, H-central series in A. Since H/A is nilpotent, it follows that H is a hypercentral group. ∎

Notice that the Fitting subgroup and the Hirsch–Plotkin radical are in general different even for metabelian Černikov groups, as the locally dihedral 2-group shows.

### The nilpotent supplementation theorem

Our next aim is to establish the existence of a nilpotent supplement for the nilpotent residual of a metanilpotent group with min-n. First we note a useful fact about the centre.

### Lemma 4.4.

Let G be a metanilpotent group with min-n, and set A=γ(G). Then AZ(G) is finite.

### Proof.

First we recall a result of Baer [3] (see also [16, Theorem 5.22]): the hypercentre of a group with min-n is a Černikov group. Thus it suffices to prove that AZ(G) has finite exponent. Assume for the moment that A is abelian, and suppose that it is unbounded. Then Corollary 3.7 shows that A has a bounded G/A-submodule B such that A/B is the direct sum of p-adically irreducible submodules Ui/B, i=1,2,,r, for various primes p. Since A=[A,G], no Ui/B can be G/A-trivial, so each CUi/B(G) is bounded and hence so is CA/B(G). Therefore, (AZ(G))B/B is bounded, and AZ(G) has finite exponent.

Returning to the general case, we know that (AZ(G))A/A has finite exponent. Since A has finite exponent by Lemma 3.5, it follows that AZ(G) has finite exponent. ∎

Next we establish the existence of hypercentral supplements.

### Lemma 4.5.

Let G be a group with min-n, and assume G has a normal nilpotent subgroup A such that G/A is hypercentral. Then there is a hypercentral Černikov subgroup X such that G=XA.

### Proof.

First of all suppose that A is abelian, and write A0=AZ¯(G). Since G/A is hypercentral and A is an artinian G/A-module, H2(G/A,A/A0)=0 by [17, Corollary AB]. Hence there is a subgroup X such that G=XA and XA=A0. Now X/A0G/A, so X/A0 is hypercentral and Černikov. Therefore, X is hypercentral. Since A0Z¯(G) and Z¯(G) is Černikov, it follows that X is Černikov.

Now let A have nilpotent class c>1. By induction on c, there exists YG such that G=YA and Y/Z(A) is a hypercentral Černikov group. Next Z(A) is an artinian Y-module since G=YA, so by the abelian case we have Y=XZ(A), where X is a hypercentral Černikov group. Thus G=YA=XA, and the result is proven. ∎

We come now to the principal supplementation theorem.

### Theorem 4.6.

Let G be a metanilpotent group with min-n, and write A=γ(G). Then there is a nilpotent Černikov subgroup X such that G=XA and XA is finite.

### Proof.

By Lemma 4.5, there is a hypercentral Černikov subgroup Y such that G=YA. Let D denote the maximum divisible subgroup of YA, so (YA)/D is finite. Now DY because D is characteristic in YA. Since Y satisfies min, there is an r>0 such that [D,rY]=[D,r+1Y]==C say. Then Y/C is nilpotent since Y is hypercentral, (YA)/D is finite and Y/YA is nilpotent. Also, C=[C,Y] is divisible. Therefore, by [17, Corollary CD], the subgroup Y nearly splits over C, i.e., Y=XC, where XC is finite. Hence G=YA=XA since CA. In addition, XCZs(X) for some s>0 since XCX and X is hypercentral. Finally, X/XCY/C, which is nilpotent. Therefore, X is nilpotent; of course, it is also a Černikov group.

It remains to prove that XA is finite. Let E denote the maximum divisible subgroup of XA. Then EZ(X)Z(A) by Lemma 2.4 since X and A are nilpotent. Consequently, EAZ(G). Since the latter is finite by Lemma 4.4, it follows that E=1 and XA is finite. ∎

In the following section, we will see that there are sharper versions of Theorem 4.6 for groups that have no proper subgroups of finite index – see Theorem 5.7.

The behaviour of the subnormal subgroups of metanilpotent groups has been studied by McCaughan and McDougal [13]. The following result is a consequence of their work.

### Theorem 4.7.

The subnormal subgroups of a metanilpotent group with min-n have bounded defects.

A corollary of this result is that the subnormal subgroups of a metanilpotent group with min-n form a complete lattice. Indeed, if a group has bounded subnormal defects, it is easy to see that any intersection of subnormal subgroups is subnormal and also that the union of an ascending chain of subnormals is subnormal. The latter property is known to imply that the join of an arbitrary set of subnormals is subnormal – for detailed proofs, see [18, 13.1].

Theorem 4.7 will follow from [13, Theorem B] once the following result has been established.

### Lemma 4.8.

Let G be a metanilpotent group with min-n, and let D be the maximum divisible subgroup of A=γ(G). If DEG, then the Sylow p-subgroups of G/E are nilpotent for all pπ(G).

### Proof.

By [7, Lemma 2.3.9], it is sufficient to prove the result for G/D; thus we can assume that D=1, so that A has finite exponent. Let PSylp(G); then ApP. If P/Ap is nilpotent, then P will be nilpotent. Thus we can assume that Ap is abelian. By Lemma 3.3, the group P/CP(Ap) is finite, so P acts nilpotently on Ap. Finally, P/ApPA/A is nilpotent, which implies that P is nilpotent. ∎

## 5 Groups without proper subgroups of finite index

If G is a metanilpotent group with min-n, it has a unique smallest subgroup of finite index, say G0. Then G0G, and G0 has no proper subgroups of finite index; moreover, G0 satisfies min-n by Wilson’s theorem. The subgroup G0 is evidently of importance for the structure of G and, remarkably, it has much simpler structure than G does. First we note a simple criterion.

## Lemma 5.1.

A metanilpotent group G with min-n has no proper subgroups of finite index if and only if G/γ(G) is a divisible abelian group.

## Proof.

Let A=γ(G), and assume that G/A is divisible abelian. Let HG have finite index, and put B=A(AH). Then A/B is finite, so G/CG(A/B) is finite and divisible. Therefore, A/B is central in G. Since A=[A,G], it follows that A=B, and hence AH. Thus H=G, which proves the sufficiency of the condition, while necessity is evident. ∎

The next result provides some basic information on the structure of the groups under consideration.

## Lemma 5.2.

Let G be a metanilpotent group with min-n which has no proper subgroups of finite index. Then

1. Z(G)=Z¯(G);

2. γ(G)=G, and G/CG((G/G′′)p) is a divisible abelian p-group for all pπ(G).

## Proof.

(i) Put Y=Z2(G) and Z=Z(G), noting that Y is a Černikov group. Then G¯=G/CG(Y) is isomorphic with a periodic subgroup of

H=Hom(Y/Z,Z).

Let D/Z denote the maximum divisible subgroup of Y/Z. If θH has finite order m, then (D/Z)θ=(D/Z)mθ=1. Hence G¯ is isomorphic with a subgroup of Hom(Y/D,Z), a group which is finite since Y/D is finite and Z is Černikov. Hence G¯ is finite, and consequently G=CG(Y). Therefore,

Y=ZandZ¯(G)=Z(G).

(ii) Let A=γ(G); then G/A is abelian, so A=G. Since G/CG((Aab)p) is divisible, it is a p-group by Lemma 3.3. ∎

Next we present some module theoretic results which are essential for further progress. The first of these is surely well known.

## Lemma 5.3.

Let p be a prime, Q a centre-by-finite p-group and A a ZpQ-module. Then AQ=0 if and only if AQ=0.

## Proof.

Assume first that AQ=0. Now we have A=[A,Q]B for some Q-submodule B by a result of Kovács and Newman [11]. Thus B𝑄A/[A,Q], so that BAQ=0 and A=[A,Q], i.e., AQ=0. Conversely, let AQ=0; then, for some submodule C, A=AQC. Hence A=[A,Q]C, whence A=C and AQ=0. ∎

Of greater consequence for our purposes here is the next result. In this, all tensor products are modules via the diagonal action of the group. We follow the notation of Section 3.

## Proposition 5.4.

Let p be a prime and Q a divisible abelian p-group, and let θ,ϕ:Q(Z¯p) be homomorphisms which are not both zero. Then

(Vθ(m)Vϕ(n))Q=0=(Vθ(m)Vϕ(n))Q

for all m,nN{}.

## Proof.

Assume that θ0. Write A=Vθ(m) and B=Vϕ(n). Then A and B have ascending composition series 0=A0A1 and 0=B0B1 with factors that are Q-isomorphic with Fθ and Fϕ respectively. Moreover, Fθ is not a trivial Q-module since θ0. Now it is straightforward to show that the Q-module AB has an ascending series whose factors are images of the modules (Ar+1/Ar)(Bs+1/Bs). Therefore, in view of Lemma 5.3, it is sufficient to prove that (FθpFϕ)Q=0.

Assume that 0z(FθpFϕ)Q. Then z is a p-linear combination of elements of the form fθgϕ, where f,gQ. (Recall that Fθ is the additive subgroup of ¯p generated by Im(θ).) All these f,g are contained in a finite subgroup H of Q such that Hθ1. Now Hθ is cyclic, say Hθ=uθ, where uH. It follows that fθuθ for all f involved in z, showing that z is a linear combination of elements of the form (uθ)r(gϕ), where 0r<|uθ|, gH.

Choose a prime q which divides m=|uθ|=|Hθ|. Hence qπ(Q) and qp. Next choose elements wiQ such that

w1q=u,wi+1q=wi,i=1,2,,

which is possible since Q is divisible. Observe that |wiθ|=mqi.

Let ei denote the exponent of p modulo |wiθ|; thus

(p(wiθ):p)=eiandpei1modmqi.

Clearly, eiei+1 and ei+1qei, so that ei+1=ei or qei. Since limiei=, there must exist an i such that ei+1=qei; fix such an i. The elements (wi+1θ)r, 0r<ei+1, are linearly independent, so they form part of a p-basis of ¯p, say ={xssS}. Next if 0r<ei, then

qr+1q(ei-1)+1=qei-(q-1)<qei=ei+1.

Now the tensors xsxt, s,tS, form a basis ¯ of ¯p¯p, and thus the (wi+1θ)qr+1xs are distinct elements of ¯ provided that 0r<ei.

Since uθwiθ, we can write

z=r,gr,g((wiθ)r(gϕ)),

where r,gp, 0r<ei and gH. Hence

zwi+1=r,gr,g((wiθ)rwi+1θ(gϕ)wi+1ϕ)=r,gr,g((wi+1θ)qr+1(gwi+1)ϕ)

since wiθ=(wi+1θ)q.

Consequently, zwi+1 is a linear combination of tensors of the form

(wi+1θ)qr+1xt,where 0r<ei,tS.

These elements belong to the basis ¯. On the other hand, we can also write

z=r,tar,t((wi+1θ)qrxt)

with ar,tp, 0r<ei, tS. Moreover, ar,t0 for some (r,t) since z0. However, the element

(wi+1θ)qrxt¯

cannot occur in the expression for zwi+1; for if it did, we would have

(wi+1θ)qr=(wi+1θ)qr+1

for some r, whence (wi+1θ)q(r-r)+1=1, and therefore (uθ)q(r-r)+1=1. But then qmq(r-r)+1, which is impossible. As a consequence, (wi+1θ)qrxt does not occur in zwi+1, and thus zzwi+1, a final contradiction. ∎

On the basis of Proposition 5.4, we obtain an important piece of structural information.

## Lemma 5.5.

Let G be a metanilpotent group with min-n, and assume that G has no proper subgroups of finite index. Write A=γ(G)=G and Q=G/A. Then

(γi(A)/γi+1(A))Q=0=(γi(A)/γi+1(A))Qfor alli.

## Proof.

We may assume that A is a p-group. Then Q¯=Q/CQ(Aab) is a divisible abelian p-group by Lemma 3.3. Put Fi=γi(A)/γi+1(A), which is a Q¯-module. Since A=[A,G], we have (Aab)Q¯=0. Assume inductively that (Fi)Q¯=0. The assignment xγi+1(A)yA[x,y]γi+2(A), where xγi(A), yA, determines a well-defined surjective Q¯-homomorphism from FiAab to Fi+1. Now Theorem 3.2 shows that Aab and Fi are direct sums of finitely many uniserial artinian Q¯-modules of the form Vθ(j) with non-zero θ since Q¯ is a p-group. Therefore, (FiAab)Q¯=0 by Proposition 5.4, and finally (Fi+1)Q¯=0=(Fi+1)Q¯ by Lemma 5.3. ∎

## Corollary 5.6.

AZ(G)=1.

It is now possible to establish an important splitting property of metanilpotent groups with min-n which have no proper subgroups of finite index.

## Theorem 5.7.

Let G be a metanilpotent group with min-n, and assume that G has no proper subgroups of finite index. Let A=γ(G)=G. Then there is a divisible abelian Černikov subgroup D such that G=DA and DA=1. Moreover, all complements of A in G are conjugate to D.

## Proof.

Let A have nilpotent class c>0, and write Q=G/A. Put B=γc(A), and assume inductively that G/B splits conjugately over A/B, say G=XA, where XA=BG. Thus X/BQ, and B is an artinian X-module. By Lemma 5.5, we have BX=BQ=0, so that H1(X/B,B)=0=H2(X/B,B) by [17, Corollary AB]. Therefore, we have X=DB, where DQ. Also, G=XA=DA, and clearly DA=1.

Suppose next that E is another complement of A in G. Now

G/B=EB/BA/Band alsoG/B=X/BA/B.

Hence, by conjugacy in G/B, we conclude that EB is conjugate to X, and we may assume that in fact X=EB. Hence X=EB=DB, and we can conclude that E is conjugate to D since H1(X/B,B)=0. ∎

The foregoing theorem is a generalization of results of McDougall [14, Theorem 5.6] and Silcock [19, Theorem B]. In fact, McDougall established the result for metabelian groups, a case of particular interest since G is the direct sum of finitely many G/CG(Ap)-modules of types Vλ(n),Vλ(), each one arising from a non-trivial simple module since A=[A,G]. This is effectively a classification of metabelian groups with min-n having no proper subgroups of finite index.

## 6 The Schur multiplier

There is a nice application of the results of the last section to Schur multipliers.

## Theorem 6.1.

Let G be a metanilpotent group with min-n, and let R denote its finite residual. Then the Schur multiplier M(G)=H2(G) is finite and has exponent dividing |G:R|2. In particular, if G has no proper subgroups of finite index, then M(G)=0.

In the proof, we will make use of the Lyndon–Hochschild–Serre spectral sequence for homology which is associated with a group extension.

## Proof of Theorem 6.1.

First of all, assume that G=R, i.e., G has no proper subgroups of finite index. Write A=γ(G), Z=Z(A), Q=G/A. Let c denote the nilpotent class of A. If c=0, then G a divisible abelian Černikov group, and thus M(G)GG=0. We proceed by induction on c>0, so that M(G/Z)=0. Note that Zp is an artinian module over Q(p)=Q/CQ(Zp) and that Q(p) is a divisible abelian p-group for pπ(Z).

Consider the homology spectral sequence for the extension ZGG/Z with coefficients in the trivial module . The relevant terms for M(G) on the E2-page are E20, E11 and E02. In the first place, E20=M(G/Z)=0. We have ZQ=0 by Lemma 5.5, which means that we can write Zp=i=1rpLi, where Li a uniserial artinian Q(p)-module of some type Vθ(n) with θ:Q(p)(¯p) non-zero.

Next, by the Künneth formula [22, Section II, 5],

M(Zp)i=1rpM(Li)i<j=1rp(LiLj).

Now M(Li)LiLi, which is a quotient of LiLi. Therefore, Proposition 5.4 applies to show that (M(Zp))Q=0 for all pπ(Z), and since

M(Z)pπ(Z)M(Zp),

it follows that E02=(M(Z))Q=0.

Finally, consider E11=H1(G/Z,Z). Using the spectral sequence for the extension A/ZG/ZQ, we find from the E2-page for this extension that E10=H1(Q,Z)=0 since ZQ=0 and Q is nilpotent. Moreover,

E01=(H1(A/Z,Z))Q(Z(A/Z)ab)Q.

By Lemma 5.5, both (A/Z)ab and Z are direct sums of finitely many Vθ(n)’s with non-zero θ. Thus, arguing as above, using Proposition 5.4, we obtain E01=0. Therefore, for the first extension, we have E11=0 and hence M(G)=0.

We return to the general case and apply the spectral sequence associated with the extension RGG/R. We have E02=0 because M(R)=0 by the first part of the proof. Further, E20=M(G/R) is finite with exponent dividing e=|G:R|. Finally, consider E11=H1(G/R,Rab); this has finite exponent dividing e. But, in addition, Rab is a Černikov group since R has min-n, and it is a routine argument to show that H1(G/R,Rab) is actually finite. It follows that M(G) is finite with exponent dividing e2. ∎

For example, the original example of Čarin has zero multiplier, as does the group obtained by using the corresponding injective module.

## Corollary 6.2.

A metanilpotent group G with min-n has finitely many covering groups.

For the isomorphism classes of covering groups of G correspond bijectively to the elements of Ext(Gab,M(G)) by [22, V, Proposition 5.3], and this is clearly finite.

## 7 Sylow properties

Metanilpotent groups with min-n, unlike periodic metabelian groups in general, have excellent Sylow properties. The standard reference for the Sylow theory of locally finite groups is Dixon [7], and we follow the terminology used there. If π is a non-empty set of primes, a Sylow π-subgroup of a group is a maximal π-subgroup. A group is said to be Sylow π-connected if all its Sylow π-subgroups are conjugate and Sylow π-integrated if every subgroup is Sylow π-connected. The following result was established by McDougall [14] in the case of metabelian groups.

## Theorem 7.1.

A metanilpotent group with min-n is Sylow π-integrated for every non-empty set of primes π.

## Proof.

Let G be a metanilpotent group with min-n, and write A=γ(G). It is sufficient to prove that G/Aπ is Sylow π-integrated. For suppose this is true. Let SG, and let P,QSylπ(S). Then SAπPQ since SAπS. By assumption, S/SAπ is Sylow π-connected. Hence P/SAπ and Q/SAπ are conjugate in S, whence so are P and Q. From now on, we will assume that Aπ=1, so A is a π-group.

Since G/A is a Černikov group, there exists AHG such that G/H is finite and H/A is abelian. Assume it has been shown that H is Sylow π-integrated, and let SG. Then SH is Sylow π-connected, while S/SH is a finite soluble group and hence is Sylow π-connected by the well-known theorem of P. Hall. If PSylπ(SH), then PA=1, so PPA/A and P is nilpotent. It now follows via [7, Theorem 2.4.4] that S is Sylow π-connected; hence G is Sylow π-integrated. This observation allows us to assume that G/A is abelian.

Let c>0 be the nilpotent class of A, and write Z=Z(A); then G/Z is Sylow π-integrated by induction on c. Let SG, and write Y=SZ; then S/Y is Sylow π-connected. Choose a Sylow π-subgroup of S/Y; by [7, Lemma 2.3.9], this has the form PY/Y for some PSylπ(S), which will be fixed from now on. Let Q be any Sylow π-subgroup of S. Now (QY/Y)sPY/Y for some sS, and hence QsPY. On replacing Q by Qs, we obtain QPY and QY=(P(QY))Y.

Since G/A is countable and abelian, there is an ascending chain

A=F0F1

of normal subgroups of G such that QA=i=1,2,Fi and Fi/A is finite. Then CZ(Fi)G. Now Z is an artinian G-module, so there exists i0 such that CZ(Fi)=CZ(Fi+1)=, and hence CZ(Fi)=CZ(QA)=CZ(Q). Moreover, Fi=(FiQ)A, which shows that Fi=FA, where F is some finite subgroup of FiQ. It follows that CZ(Q)=CZ(F) and, on intersecting with S, we obtain CY(Q)=CY(F). Apply [7, Lemma 2.4.6] to the semidirect product

QY=(P(QY))Y,

keeping in mind that A is a π-group. Hence all complements of Y in QY are conjugate, so Qx=P(QY)P for some xQYS. Since QxSylπ(S), we deduce that Qx=P. Hence S is Sylow π-connected and G is Sylow π-integrated. ∎

On the other hand, as Dixon [7, Example 4.5.2] has observed, there are soluble groups with min-n of derived length 3 which are not Sylow p-connected for some prime p.

## Corollary 7.2.

A metanilpotent group with min-n has countably many Sylow π-subgroups for all π.

This is a consequence of Theorem 7.1 and the countability of the groups concerned. The next result, which generalizes work of Silcock [19, Theorem A], gives information on the structure of the Sylow p-subgroups in situations where there are no proper subgroups of finite index.

## Corollary 7.3.

Let G be a metanilpotent group with min-n which has no proper subgroups of finite index. Then the following hold.

1. Each Sylow p-subgroup of G is the direct product of a divisible abelian p-group of finite rank and (G)p.

2. For pπ(G), the Sylow p-subgroups of G are nilpotent with the same class as (G)p.

## Proof.

By Theorem 5.7, we can write G=DA, where A=G and D is a divisible abelian group of finite rank. Clearly, P=DpAp is a Sylow p-subgroup of G. By Lemma 3.3, the subgroup Dp centralizes (Ap)ab, so P/(Ap), and hence P, is nilpotent. Therefore, DpZ(P) by Lemma 2.4; also, P=Dp×Ap and γi(P)=γi(Ap) for i2. Hence P and Ap have the same nilpotent class for pπ(A). All the Sylow p-subgroups have this form since they are conjugate in G. ∎

In particular, the Sylow p-subgroups of a metabelian group with min-n and no proper subgroups of finite index are abelian, which is a result of McDougall [14, Theorem 3.2].

## 8 Frattini subgroups

It is easy to find metanilpotent groups with min-n which have non-nilpotent Frattini subgroups, even in the Černikov case.

1. Let G=x(AA), where A is a 2-group, x4=1 and

(a1,a2)x=(a2,-a1),aiA;

here ϕ(G)=x2,AA, which is not nilpotent.

2. Let p,q be distinct primes, and let A be an injective uniserial Q-module arising from a simple module of Čarin type (p,q), where Q is a q-group. Let G=QA; in this case, ϕ(G)=G, which is not even locally nilpotent.

These examples suggest that the obstruction to nilpotence of the Frattini subgroup of a metanilpotent group G with min-n may lie in the divisible part of (γ(G))ab. This is confirmed by the following theorem.

## Theorem 8.1.

Let G be a metanilpotent group with min-n, and put A=γ(G). Let D/A denote the maximum divisible subgroup of Aab, and let N be a normal subgroup of G containing ϕ(G). Then

1. N is locally nilpotent if and only if N/ϕ(G) is locally nilpotent and N/D acts hypercentrally on D/A,

2. N is nilpotent if and only if N/ϕ(G) is nilpotent and N/D centralizes D/A.

Since A is nilpotent, Aϕ(G). Also, if M is a maximal subgroup of G that does not contain A, then A/AM is a simple G-module and hence is elementary abelian. Thus DAM, so that we have ADϕ(G)N and ϕ(G/D)=ϕ(G)/D. We can take N to be ϕ(G) in Theorem 8.1, thus obtaining the next result.

## Corollary 8.2.

The following statements hold.

1. ϕ(G) is locally nilpotent if and only if ϕ(G/D) acts hypercentrally on D/A.

2. ϕ(G) is nilpotent if and only if ϕ(G/D) centralizes D/A.

Recall the well-known theorem of W. Gaschütz that, for any finite group H,

Fitt(H/ϕ(H))=Fitt(H)/ϕ(H).

There is an analogue of this result for metanilpotent groups with min-n; in what follows, we maintain the notation of the theorem.

## Corollary 8.3.

The following statements hold.

1. Assume that ϕ(G) is locally nilpotent, and put L/ϕ(G)=HP(G/ϕ(G)). Then HP(G/ϕ(G))=HP(G)/ϕ(G) if and only if L/D acts hypercentrally on D/A.

2. Assume that ϕ(G) is nilpotent, and put M/ϕ(G)=Fitt(G/ϕ(G)). Then Fitt(G/ϕ(G))=Fitt(G)/ϕ(G) if and only if M/D centralizes D/A.

To prove Corollary 8.3, take N to be successively L and M in Theorem 8.1.

## Proof of Theorem 8.1.

(i) In the first place, if N is locally nilpotent, it is hypercentral by Theorem 4.3. Thus the conditions in (i) are necessary. Assume that N satisfies the conditions, and observe that these are inherited by quotients of G. If the result has been proved for G/A, then NA/A is locally nilpotent, which implies that NA/A is locally nilpotent by the Hirsch–Plotkin theorem; hence N is locally nilpotent ([15, Theorem 3]). Therefore, we can assume that A is abelian. We may also assume that A is a p-group; for if NAp/Ap is locally nilpotent for all pπ(A), then N is locally nilpotent.

Put A0=AZ¯(G); if NA0/A0 is locally nilpotent, then N is locally nilpotent, an observation that allows us to assume that AZ¯(G)=1. Thus, by [17, Corollary AB], H2(G/A,A)=0, and hence G splits over A, say G=QA, where Q is a nilpotent Černikov group. Write Q=P×R, where P and R are the p- and p-components of Q. Using bars to denote quotients modulo CQ(A), we have Q¯=P¯×R¯, and P¯ is a finite p-group by Lemma 3.3.

For the moment, suppose that Ap=1. The pR¯-module A/[A,P] is artinian, and hence it is the direct sum of finitely many simple pR¯-modules Bi/[A,P], i=1,2,,r (by Theorem 3.2, for example). Put Ci=j=1,2,r,jiBj. Then A/Ci is a simple module, and thus QCi is a maximal subgroup of G. Therefore,

ϕ(G)i=1rQCi=Qi=1rCi=Q[A,P]

and ϕ(G)A[A,P].

Now let U be a finitely generated subgroup of N; then Uϕ(G)/ϕ(G) is nilpotent, and γn(U)ϕ(G)A[A,P] for some n>0. Hence U[A,P]/[A,P] is nilpotent, and N[A,P]/[A,P] is locally nilpotent. Moreover, PA is nilpotent since Ap=1, PA/A is nilpotent and P/CP(A)P¯ is a finite p-group. Since PAQA=G, it follows via the Hirsch–Plotkin theorem that NPA/[A,P] is locally nilpotent. Also, (PA)=P[A,P], so that NPA/(PA) is locally nilpotent, which implies that N is locally nilpotent.

Returning to the general case, from the previous discussion, we know that NAp/Ap is locally nilpotent, and hence so is NA/Ap. This means that N acts hypercentrally on A/Ap, so it acts hypercentrally on each factor Api/Api+1. However, Apn=Apn+1=D for some n>0, so N acts hypercentrally on A/D. By hypothesis, N acts hypercentrally on D and hence on A. Since NA/A is nilpotent, NA is a hypercentral group, as is N, which establishes (i).

(ii) First of all, assume that N is nilpotent. Then NA is nilpotent. Since D/A is divisible, it follows from Lemma 2.4 that N centralizes D/A. Conversely, assume that the conditions on N in (ii) hold. As before, we may assume that A is abelian. By (i), the subgroup N – hence NA – is hypercentral, so N acts hypercentrally on A/D. Proposition 3.4 shows that the G/A-module A/D is a noetherian, from which it follows that N acts nilpotently on A/D. Since, by hypothesis, N centralizes D, it follows that N acts nilpotently on A. Therefore, N is nilpotent. ∎

## 9 Maximal subgroups

The maximal subgroups of metanilpotent groups satisfying min-n are objects that are worthy of study, as is suggested by results in Section 8. In particular, questions about their cardinality merit attention; one notable fact that seems to have escaped notice by earlier researchers is that there are just countably many of them.

## Theorem 9.1.

A metanilpotent group with min-n has countably many maximal subgroups.

A generalization of this result appears in [1, Theorem 7], where many other classes of groups with a countable number of maximal subgroups are identified. We include a short proof of the theorem.

## Proof of Theorem 9.1.

Let G be a metanilpotent group satisfying min-n, and put A=γ(G); then A is nilpotent, so Aϕ(G), and we can assume that A is abelian. Since G/A is a Černikov group, it has finitely many maximal subgroups.

Assume the result is false; then there are uncountably many maximal subgroups M not containing A. For such an M, we have G=MA and MAG, while MA is a maximal G-submodule of A. Let B denote the intersection of all the maximal submodules of A. Since A is artinian, B is the intersection of finitely many maximal submodules, and thus A/B is noetherian. We can assume that B=1. Since A is noetherian, it has only countably many submodules, so there must exist uncountably many maximal subgroups M such that G=MA and MA=A0 is fixed. By passing to G/A0, we may assume that MA=1 for all such M. Thus G=MA, and A is a simple G-module.

Next CA(G)=1 since A is simple and 1<A=[A,G]. By [17, Corollary AB], we have H1(G/A,A)=0 and hence Der(G/A,A)=Inn(G/A,A). Since A is countable, so is Inn(G/A,A). It follows that there are only countably many complements of A in G and hence countably many subgroups M. This is a contradiction. ∎

We consider next the stronger property – for countable groups at least – that there are finitely many conjugacy classes of maximal subgroups. As it turns out, not all metanilpotent groups with min-n possess this property.

## Theorem 9.2.

Let G be a metanilpotent group with min-n, and write A=γ(G) and Q=G/A. Then the following statements are equivalent.

1. The maximal subgroups of G fall into finitely many conjugacy classes.

2. No two infinite simple Q-quotients of Aab are Q-isomorphic.

## Proof.

If D/A denotes the maximum divisible subgroup of Aab, then Dϕ(G). Therefore, we can factor out by D and assume that A is an abelian group of finite exponent. Hence A has finite Q-composition length by Proposition 3.4.

(i) (ii) Assume that (i) is valid, but nevertheless there exist distinct isomorphic infinite simple Q-quotients A/A1 and A/A2. Then A=A2+A1 and A/A1A2𝑄(A/A1)(A/A2). Thus, in order to reach a contradiction, we may assume that A=BC, where B,C are isomorphic infinite simple Q-modules. Observe that B and C are non-trivial as Q-modules since A=[A,G]. Consequently, CA(G)=1 and H2(Q,A)=0; thus GQA.

Now fix B, and consider the Q-module complements of B in A. These correspond to elements of HomQ(A/B,B)EndQ(B). Let E denote the maximum divisible subgroup of the Černikov group Q. If [B,E]=1, then [A,E]=1, and hence B and C are finite since they are simple Q/E-modules. By this contradiction, [B,E]1, and hence E/CE(B) is infinite. Let eE\CE(B); then, since EZ(Q), the mapping τe defined by b[b,e],(bB), belongs to EndQ(B). Also τe1=τe2 implies that e1e2-1CE(B). It follows that EndQ(B) is infinite, which means that there are infinitely many Q-complements of B in A. If C1 and C2 are two such complements, then QCi is maximal in G. Moreover, QC1 and QC2 are clearly not conjugate in G, which contradicts (i).

(ii) (i) Assume that condition (ii) holds, yet G has infinitely many pairwise non-conjugate, maximal subgroups. Let A0 denote the minimum Q-submodule with finite index in A. Then A/A0 is finite, and G/A0 is a Černikov group. Hence only finitely many maximal subgroups of G can contain A0. Consequently, there is an infinite set of mutually non-conjugate, maximal subgroups of G that fail to contain A0. If M, then G=MA and A/MA is an infinite simple Q-module. Each A/MA is isomorphic with a factor in a fixed Q-composition series of A, so by condition (ii) there is an infinite subset 0 of such that MA=L is fixed for all M0. Notice that A/L is a non-trivial simple Q-module; therefore, H1(G/A,A/L)=0, showing that all the complements of A/L in G/L=M/LA/L are conjugate and hence that all M0 are conjugate; this is a contradiction. ∎

The second condition in Theorem 9.2 is clarified by a simple observation. In what follows, ϕQ(A) denotes the intersection of all the maximal submodules of the Q-module A.

## Lemma 9.3.

Let A be an artinian module over a nilpotent Černikov group Q. Then the following conditions are equivalent.

1. No two infinite simple quotient modules of A are isomorphic.

2. A/ϕQ(A) is the direct sum of finitely many non-isomorphic simple modules.

## Proof.

There is no loss in assuming that ϕQ(A)=0. Also, it is straightforward to show that we can assume A to be a p-group. Thus A is an elementary abelian p-group.

(i) (ii) Let Q¯=Q/CQ(A); then we can write Q¯=P¯×R¯, where P¯ is a finite p-group and R¯ is a p-group. Let A/B be a simple Q¯-quotient; then P¯(A/B) is nilpotent, so [A,P¯]B for all such B. Therefore, [A,P¯]=0 and Q¯=R¯. By Theorem 3.2, the module A is the direct sum of finitely many simple Q¯-modules. By condition (i), the isomorphism types of the factors in the direct decomposition are different, so (ii) holds.

(ii) (i) Let A=i=1kAi be the given decomposition, with Ai a simple module of type λi. Consider a simple Q-quotient A/B of A. Then AjB for some j and A=AjB. Hence A/B is a simple module of type λj. Since the λi are all different, we must have AiB for all ij, and thus B=ij,i=1kAi. Hence B is determined by the isomorphism type of A/B, which establishes the validity of (i). ∎

A still more stringent group theoretic property is that of having finitely many maximal subgroups. It is easy to see that a group has finitely many maximal subgroups if and only if the Frattini subgroup has finite index. For groups in our class, the situation is clarified by the following result.

## Theorem 9.4.

Let G be a metanilpotent group with min-n. Then the following conditions are equivalent.

1. G has finitely many maximal subgroups.

2. (γ(G))ab has a divisible subgroup of finite index.

3. G is an extension of a divisible abelian normal subgroup by a Černikov group.

## Proof.

Write A=γ(G), noting that Aϕ(G).

(i) (ii) We may assume that A is abelian; write D for its maximum divisible subgroup. Then Dϕ(G) and ϕ(G/D)=ϕ(G)/D. Hence ϕ(G)/D is nilpotent by Corollary 8.2. Moreover, as G/ϕ(G) is finite, ϕ(G) has min-n, so G/D is Černikov. Since A/D has finite exponent, it must be finite.

(ii) (iii) Define E=n=1,2,An!. Then, by Lemma 2.4, EZ(A), and E is divisible, while A/E has finite exponent. Thus E is the maximum divisible subgroup of A, and it follows from (ii) that A/E is finite. Therefore, G/E is a Černikov group.

(iii) (i) This is clear since a maximal subgroup contains the divisible abelian normal subgroup. ∎

## Corollary 9.5.

Let G be a metanilpotent group with min-n. Then G has no maximal subgroups if and only if G has the form DA, where D and A are divisible abelian groups and D is a Černikov group.

## Proof.

Assume that G has no maximal subgroups. Then G has no proper subgroups of finite index, and thus A=γ(G)=G by Lemma 5.2. Let A0 denote the minimum G-invariant subgroup with finite index in A. Thus G/A0 is a Černikov group, so it is abelian, which implies that A0=A. Since A is nilpotent, it follows from Lemma 2.4 that A is abelian and divisible. The assertion is now a consequence of Theorem 5.7. The converse is obvious. ∎

## 10 Countable domination

Quite recently, there has been interest in metanilpotent groups with min-n that was motivated by research on countability restrictions on subgroup lattices. One such restriction is that of countable domination.

A group G is said to be countably dominated (or CD) if there is a countable set 𝒮 of proper subgroups such that every proper subgroup of G is contained in some member of 𝒮. If G is a CD-group, clearly it can have only countably many maximal subgroups. Thus, in the light of Theorem 9.1, it is natural to ask whether metanilpotent groups with min-n are countably dominated. This turns out to be a subtle question whose answer is also module theoretic in character. The following definitive result is established in [2, Theorem 5.1].

## Theorem 10.1.

Let G be a metanilpotent group with min-n, and put A=γ(G), Q=G/A. Then G is countably dominated if and only if Aab has only countably many Q-submodules and the finite residual of Q is locally cyclic.

The condition on Aab in the theorem can be expressed in terms of the Q-module structure.

## Theorem 10.2 ([2, Theorem 5.7]).

Let A be an artinian module over a nilpotent Černikov group Q. Then the following statements are equivalent.

1. A has countably many submodules.

2. A=(A1A2An)+S, where the Ai are pairwise non-near isomorphic, p-adically irreducible submodules for various primes p and S is a bounded submodule.

Here the modules Ai,Aj are said to be near isomorphic if Ai/BiAj/Bj for some bounded submodules Bi and Bj.

We conclude by illustrating these results with some examples. Let p and q be distinct primes, and let Q be a q-group. Let A be a simple Q-module of Čarin type (p,q). Then the Čarin (p,q)-group G1=QA is a CD-group since γ(G1)=A. Next let B denote the injective hull of the module A, and put G2=QB; then γ(G2)=B, and G2 is also a CD-group. On the other hand, G3=Q(BB) is not a CD-group because γ(G3)=BB, which is the direct sum of two isomorphic p-adically irreducible Q-modules.

Communicated by Andrea Lucchini

## References

[1] A. Arikan, G. Cutolo and D. J. S. Robinson, On groups with countably many maximal subgroups, J. Group Theory, 21 (2018), 253–271. 10.1515/jgth-2017-0035Search in Google Scholar

[2] A. Arikan, G. Cutolo and D. J. S. Robinson, On groups that are dominated by countably many proper subgroups, J. Algebra 509 (2018), 445–466. 10.1016/j.jalgebra.2018.05.018Search in Google Scholar

[3] R. Baer, Groups with descending chain condition for normal subgroups, Duke Math. J. 16 (1949), 1–22. 10.1215/S0012-7094-49-01601-4Search in Google Scholar

[4] R. Baer, Irreducible groups of automorphisms of abelian groups, Pacific J. Math. 14 (1964), 385–406. 10.2140/pjm.1964.14.385Search in Google Scholar

[5] V. S. Čarin, A remark on the minimal condition for subgroups, Dokl. Akad. Nauk SSSR (N. S.) 66 (1949), 575–576. Search in Google Scholar

[6] S. N. Černikov, On locally soluble groups which satisfy the minimal condition for subgroups, Mat. Sb. 28 (1951), 119–129. Search in Google Scholar

[7] M. R. Dixon, Sylow Theory, Formations and Fitting Classes in Locally Finite Groups, Ser. Algebra. 2, World Scientific, River Edge, 1994. 10.1142/2386Search in Google Scholar

[8] B. Hartley, Uncountable artinian modules and uncountable soluble groups satisfying min-n, Proc. Lond. Math. Soc. (3) 35 (1977), 55–75. 10.1112/plms/s3-35.1.55Search in Google Scholar

[9] B. Hartley and D. McDougall, Injective modules and soluble groups satisfying the minimal condition for normal subgroups, Bull. Aust. Math. Soc. 4 (1971), 113–135. 10.1017/S0004972700046335Search in Google Scholar

[10] H. Heineken and J. S. Wilson, Locally soluble groups with min-n, J. Aust. Math. Soc. 17 (1974), 305–318. 10.1017/S1446788700017079Search in Google Scholar

[11] L. G. Kovács and M. F. Newman, Direct complementation in groups with operators, Arch. Math. (Basel) 13 (1962), 427–433. 10.1007/BF01650091Search in Google Scholar

[12] J. C. Lennox and D. J. S. Robinson, The Theory of Infinite Soluble Groups, Oxford University, Oxford, 2004. 10.1093/acprof:oso/9780198507284.001.0001Search in Google Scholar

[13] D. J. McCaughan and D. McDougall, The subnormal structure of metanilpotent groups, Bull. Aust. Math. Soc. 6 (1972), 287–306. 10.1017/S0004972700044506Search in Google Scholar

[14] D. McDougall, Soluble groups with the minimal condition for normal subgroups, Math. Z. 118 (1970), 157–167. 10.1007/BF01113337Search in Google Scholar

[15] D. J. S. Robinson, A property of the lower central series of a group, Math. Z. 107 (1968), 225–231. 10.1007/BF01110261Search in Google Scholar

[16] D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Springer, Berlin, 1972. 10.1007/978-3-662-07241-7Search in Google Scholar

[17] D. J. S. Robinson, The vanishing of certain homology and cohomology groups, J. Pure Appl. Algebra 7 (1976), 145–167. 10.1016/0022-4049(76)90029-3Search in Google Scholar

[18] D. J. S. Robinson, A Course in the Theory of Groups, 2nd ed., Grad. Texts in Math. 80, Springer, New York, 1995. Search in Google Scholar

[19] H. L. Silcock, Metanilpotent groups satisfying the minimal condition for normal subgroups, Math. Z. 135 (1974), 165–173. 10.1007/BF01189353Search in Google Scholar

[20] H. L. Silcock, On the construction of soluble groups satisfying the minimal condition for normal subgroups, Bull. Aust. Math. Soc. 12 (1975), 231–257. 10.1017/S0004972700023844Search in Google Scholar

[21] H. L. Silcock, Representations of metabelian groups satisfying the minimal condition for normal subgroups, Bull. Aust. Math. Soc. 14 (1976), 267–278. 10.1017/S0004972700025089Search in Google Scholar

[22] U. Stammbach, Homology in Group Theory, Lecture Notes in Math. 359, Springer, Berlin, 1973. 10.1007/BFb0067177Search in Google Scholar

[23] J. S. Wilson, Some properties of groups inherited by normal subgroups of finite index, Math. Z. 114 (1970), 19–21. 10.1007/BF01111865Search in Google Scholar