Soluble Minimax Research Articles

Potency in soluble groups

We prove in particular that if G is a soluble group with no non-trivial locally finite normal subgroups, then G is p-potent for every prime p for which G has no Prüfer p-sections. (A group G is p-potent if for every power n of p and for any element x of G of infinite order or of finite order divisible by n there is a normal subgroup N of G of finite index such that the order of x modulo N is n. A Prüfer p-group is an infinite locally cyclic p-group.) This extends to soluble groups in general, and gives a more direct proof of, recent results of Azarov on polycyclic groups and soluble minimax groups.

Sylow permutability in soluble minimax groups

A PST-group is a group in which Sylow permutability is a transitive relation. A classification is given of the soluble minimax groups whose finite quotients are PST-groups.

On the residual finiteness of the HNN-extensions and generalized free products of finite rank groups

We obtain necessary and sufficient conditions for the residual finiteness for some HNN-extensions and generalized free products of soluble minimax groups.

Groups Whose Finite Homomorphic Images are Metahamiltonian

A group G is metahamiltonian if all its non-abelian subgroups are normal. It is proved here that a finitely generated soluble group is metahamiltonian if and only if all its finite homomorphic images are metahamiltonian; the behaviour of soluble minimax groups with metahamiltonian finite homomorphic images is also investigated. Moreover, groups satisfying the minimal condition on non-metahamiltonian subgroups are described.

Generation of polycyclic groups

We give a new and self-contained proof of a theorem of Linnell and Warhurst that d(G) – d(Ĝ) ⩽ 1 for finitely generated virtually torsion-free soluble minimax groups G. We also give a simple sufficient condition for the equality d(G) = d(Ĝ) to hold when G is virtually abelian.

On modules of finite upper rank

For a group $G$ and a prime $p$, the upper $p$-rank of $G$ is the supremum of the sectional $p$-ranks of all finite quotients of $G$. It is unknown whether, for a finitely generated group $G$, these numbers can be finite but unbounded as $p$ ranges over all primes. The conjecture that this cannot happen if $G$ is soluble is reduced to an analogous ‘relative’ conjecture about the upper $p$-ranks of a ‘quasi-finitely-generated’ module $M$ for a soluble minimax group $\Gamma$. The main result establishes a special case of this relative conjecture, namely when the module $M$ is finitely generated and the minimax group $\Gamma$ is abelian-by-polycyclic. The proof depends on generalising results of Roseblade on group rings of polycyclic groups to group rings of soluble minimax groups. (If true in general, the above-stated conjecture would imply the truth of Lubotzky’s ‘Gap Conjecture’ for subgroup growth, in the case of soluble groups; the Gap Conjecture is known to be false for non-soluble groups.)

On Groups of Automorphisms of Residually Finite Groups

We show that certain properties of groups of automorphisms can be read off from the actions they induce on the finite characteristic quotients of their underlying group G. In particular, we obtain criteria for groups of automorphisms of a residually finite and (soluble minimax)-by-finite group G to be nilpotent or soluble. Moreover, we give explicit bounds on the class (the derived length, resp.) of such groups of automorphisms in terms of invariants of G. Finally, we consider similar questions when G is the free group of rank two.

Linear Bounds for the Degree of Subgroup Growth in Terms of the Hirsch Length

Let G be a residually-finite virtually soluble minimax group. We give upper and lower linear bounds for the degree of subgroup growth of G in terms of the Hirsch length of G .

Groups without nearly abnormal subgroups

A subgroup M of an infinite group G is said to be nearly maximal if it is a maximal element of the set of all subgroups of G having infinite index; i.e. if the index |G:M| is infinite but every subgroup of G properly containing M has finite index in G. The near Frattini subgroup ψ(G) of an infinite group G can now be defined as the intersection of all nearly maximal subgroups of G, with the stipulation that ψ(G)=G if G has no nearly maximal subgroups. These concepts have been introduced by Riles [5]. It was later proved by Lennox and Robinson [4] that a finitely generated soluble-by-finite group G is infinite-by-nilpotent if and only if all its nearly maximal subgroups are normal. It follows that in the class of finitely generated soluble-by-finite groups the property of being finite-by-nilpotent is inherited from the near Frattini factor group G/ψ(G) to the group G itself. In the study of ordinary Frattini properties of infinite groups, some analogies exist between the behaviour of finitely generated soluble groups and soluble minimax residually finite groups (see for instance [6] and [7]). This fact could suggest that a result corresponding to that of Lennox and Robinson also holds for soluble residually finite minimax groups. Unfortunately in this case the property of being finite-by-nilpotent cannot be detected from the behaviour of nearly maximal subgroups, this phenomenon depending on the fact that infinite soluble residually finite minimax groups may be poor of such subgroups.

The Permutizer Condition in Infinite Soluble Groups

A group satisfies the permutizer conditionPif each proper subgroup permutes with some cyclic subgroup not contained in it. Here we characterize the classes of soluble minimax groups and finitely generated soluble groups withP.

On soluble groups with finite homomorphic images of bounded rank

Let G be a group. We denote by 9r(G) the set of all finite homomorphic images of G. The study of the connection between the properties of G and of groups of the set ~'(G) is a traditional problem of group theory. In this connection the following question, proposed by Wilson (see [1, Question 12.90]), is of interest: is a finitely generated residually finite group G minimax if there exists a finitely generated soluble minimax group H such that 9V(G) = 9V(H)? In the present paper we show that G is also minimax if ~-(G) C_ ~'(H). We recall that a group is minimax if it has a finite subnormal series each factor of which satisfies either the minimality condition or the maximality condition for subgroups. The basic properties of minimax groups are discussed in [2]. Lemma 1. Let G be a residually finite group, and let H be a group of finite rank. If ~(G) C ~(H), then G does not contain infinite elementary Abelian p-subgroups. Proof. Suppose that G contains an infinite elementary Abelian p-subgroup A. Then G contains a finite elementary Abelian p-subgroup B such that [B I > pr, where r is the rank of H. Since G is a residually finite group, we see that G contains a normal subgroup R of finite index such that B fq R = 1. Then G = G/R contains an elementary Abelian p-subgroup/~ such that IBI > p~. Since G E 37(G) and ~-(G) C ~-(g), we have G E 5r(g). Therefore, r(G) pL The lemma has been proved. Lemma 2. Let G be a subgroup of the Cartesian product G = :~iGi of finite soluble groups Gi of bounded orders. Suppose that H is a group of finite rank. If ~(G) C 3U(H), then G is a finite group. Proof. Since the orders of Gi are bounded, it follows that the solubility lengths of Gi are also bounded and, hence, G is soluble. Suppose that G is an infinite group and show by induction on the solubility length of G that the latter contains an infinite elementary Abelian p-subgroup. Assume that G is an Abelian group. Since there exists n E N such that IGil < n for all i, it follows that there exists t E N such that G~ = 1 for all i; therefore, G t = 1. From this it follows that G contains an infinite elementary Abelian p-subgroup. Let D be a derived subgroup of G. If ]D I = :xD, then by the inductive hypothesis G contains an infinite elementary Abelian p-subgroup. Thus, one can assume that IDI < :xD. Since G is residually finite, there exists a normal subgroup R < G of finite index such that RND = 1 and, consequently, R is an infinite Abelian subgroup of G. As was shown above, R contains an infinite elementary Abelian p-subgroup. Thus, G contains an infinite elementary Abelian p-subgroup. This contradicts Lemma 1. Therefore, the group G is finite. The lemma has been proved. Lemma 3. Let G be a residually finite group, and let H be an almost metanilpotent soluble group of finite rank. If ~(G) C F(H), then G is an almost metanilpotent group. Proof. Since G is a residually finite group, we see that G is a subgroup of the Cartesian product = ~iGi, where Gi E JZ(G). The group H has the normal series

Torsion in profinite completions

In [l] the question was raised whether the profinite completion of a torsion-free residually finite group is necessarily torsion-free. Subsequently, counter-examples were discovered by Evans [2], and more recently, Lubotsky [4] has shown that there is a finitely generated torsion-free residually finite group whose profinite completion contains a copy of every finite group. In this paper we study the situation more closely for soluble groups. The major result of the paper is the construction of finitely generated centre by metabelian counter-examples in Section 3. We begin in Section 2 by establishing some positive results showing that there are no counter-examples amongst abelian groups, finitely generated abelian by nilpotent groups or soluble minimax groups. In Section 3 we first establish counter-examples which are nilpotent of class 2: these are necessarily infinitely generated. Then we give the promised centre by metabelian examples. It is interesting to note that the dichotomy arising here between abelian by nilpotent groups and centre by metabelian groups is the one which arises in the work of Hall [3].

Fitting subgroups of profinite completions of soluble minimax groups

How far does the class of all finite images of a group determine this group? It is known that finitely generated abelian groups are determined by their finite images up to isomorphism whereas finitely generated nilpotent groups in general are not [4]. At least there are only finitely many nonisomorphic such groups with the same finite images [S]. The same is true for polycyclic-by-finite groups [3] and for finite extensions of torsion-free nilpotent minimax groups which are radicable by its spectrum [7]. It is not known if there is a common generalization to soluble minimax groups the Fitting subgroups of which are radicable by their spectrum. In the way Pickel’s theorem [9] was a first step to the proof of the result on polycyclic groups the Main Theorem of this paper might be helpful on the way of a proof of such a more general result.

Subnormality and quasinormality in soluble minimax groups

Subnormality and quasinormality in soluble minimax groups

Groups whose finite quotients are supersoluble

I f G is an infinite group each of whose finite quotients has some given property, one may ask how this situation is reflected in the global structure of the group G. The first result in this area of investigation was obtained by Hirsch in [5] where he proved that a polycyclic group G each of whose finite quotients is nilpotent must itself be nilpotent. This result has been extended in two directions by D. J. S. Robinson, who showed that the same holds if G is merely assumed to be either a torsion-free by finite soluble minimax group or a finitely generated hyper-(abelian or finite) group ([7] and [S]). On the other hand W’ehrfritz and Platonov proved (independently) that the same result holds under the assumption that G is a linear group over a finitely generated integral domain (see [ 121, 4.16 and 10.15). Now supersolubility is in some ways a subtler property than nilpotency, and knowledge of the groups in the title seems to be rather more fragmentary. Baer [l] has shown that if a group G each of whose finite quotients is supersoluble is polvcvclic, then G must itself be supersoluble; and Wehrfritz [I l] . I has shown that if G is a linear group over a finitely generated integral domain of nonzero characteristic then G is hypercyclic (he has recently generalized this in [ 131; we quote his theorem in full in Section 6). In this paper we extend the former result to larger classes of (roughly) soluble groups, and we settle the characteristic-zero case of the linear group problem.

Subnomality in soluble minimax groups

A subgroup H of a group G is said to be subnormal in G if there is a finite chain of subgroups, each normal in its successor, connecting H to G. If such chains exist there is one of minimal length; the number of strict inclusions in this chain is called the subnormal index, or defect, of H in G. The rather large class of groups which have an upper bound for the subnormal indices of their subnormal subgroups has been inverstigated to same extent, mainly with a restriction to solublegroups — for instance, in [10] McDougall considered soluble p-groups in this class. Robinson, in [14], restricted his attention to wreath products of nilpotent groups but extended his investigations to the strictly larger class of groups in which the intersection of any family of subnormal subgroups is a subnormal subgroup. These groups are said to have the subnormal intersection property.

A theorem on finitely generated hyperabelian groups

Our main object is to prove that a finitely generated non-nilpotent hyperabelian group has a finite non-nilpotent homomorphic image. The terminology "here is as follows: if 9 ~ is a group theoretical property, a group G is a hyper-~ group if it has an ascending series (of arbitrary ordinal type beginning at 1 and ending at G) of normal subgroups such that each factor of the series has the property ~: when ~ is inherited by homomorphic images, it is not difficult to see that hyper-~ is equivalent to the requirement that every non-trivial homomorphic image contain a non-trivial normal subgroup with the property ~. Actually we will prove a slight generalization of the result mentioned above. Theorem 1. Let G be a finitely generated hyper-(abelian or finite) group. If G is not nilpotent, it has a finite non-nilpotent homomorphic image. The special case where G is polycyclic is due to Hirsch ([6], Theorem 3.24). At least two generalizations of Hirsch's theorem already exist. A non-nilpotent soluble minimax group without groups of type pO~ in the centre of its Fitting subgroup for any prime p has a finite non-nilpotent homomorphic image (Robinson [7], Theorem A and Theorem 5.11). (A minimax group is a group which has a series of finite length whose factors satisfy either the maximal or minimal condition on subgroups.) A similar result is true for non-nilpotent linear groups over a finitely generated integral domain (cf. Wehrfritz [10], Lemma 2).