Finite Abelian Section Rank Research Articles

Groups with all subgroups permutable or soluble of finite rank

In this paper the authors study the class of locally graded groups all of whose subgroups are permutable or are soluble and satisfy a certain rank condition. The rank conditions in question include groups of finite abelian section rank, minimax groups and polycyclic groups. In each case necessary and sufficient conditions are given for a locally graded group to have all subgroups permutable or soluble with the given rank condition.

On Groups with Finite Abelian Section Rank Factorized by Mutually Permutable Subgroups

It is proved that if G = AB is a soluble group with finite abelian section rank which is factorized by two mutually permutable finite-by-nilpotent subgroups A and B such that A′ and B′ are locally nilpotent, then also the normal closure ⟨ A′, B′ ⟩G is locally nilpotent and the subgroups A′ and B′ are ascendant in G.

Torsion cohomology for solvable groups of finite rank

We define a class U of solvable groups of finite abelian section rank which includes all such groups that are virtually torsion-free as well as those that are finitely generated. Assume that G is a group in U and A a ZG-module. If A is Z-torsion-free and has finite Z-rank, we stipulate a condition on A that guarantees that Hn(G,A) and Hn(G,A) must be finite for n≥0. Moreover, if the underlying abelian group of A is a Černikov group, we identify a similar condition on A that ensures that Hn(G,A) must be a Černikov group for all n≥0.

‘COHOMOLOGY AND PROFINITE TOPOLOGIES FOR SOLVABLE GROUPS OF FINITE RANK’

AbstractWe remedy an omission in the proof of Proposition 2.7 of the paper ‘Cohomology and profinite topologies for solvable groups of finite rank’, Bull. Aust. Math. Soc.86 (2012), 254–265. This proposition states that a solvable group with finite abelian section rank has merely finitely many subgroups of any given index.

On Hyper (Abelian of Finite Rank) Groups

We study the class of groups G, each of whose non-trivial images contains a non-trivial abelian normal subgroup of finite rank. This is very much wider than the class, studied earlier by Robinson and others, of hyperabelian groups H with finite abelian section rank. Our main results are that these groups G are hypercentral by residually finite and are divisible by hypo(abelian of finite exponent). They need not be divisible by residually finite, unlike the groups H above. In practice, we work with a somewhat wider but less easily described class of groups.

MC-hypercentral groups

This paper is devoted to the imposition of some chain conditions on groups having a generalized central series. It is also given a characterization of MC-groups with finite abelian section rank: such class of groups is a suitable enlargement of the class of FC-groups. Mathematics Subject Classification: 20F24; 20F14

Hyper-abelian groups with finite co-central rank

A group G has finite co-central rank s if there exists a least non-negative integer s such that every finitely generated subgroup H can be generated by at most s elements modulo the centre of H. The investigation of such groups has been started in [J.P. Sysak, A. Tresch, J. Group Theory 4 (2001) 325]. It is proved that hyper-abelian groups with finite co-central rank are locally soluble. The interplay between the Prüfer rank condition, the condition of having finite abelian section rank and the finite co-central rank condition is studied. As one result, a hyper-abelian group G with finite co-central rank has an ascending series with abelian factors of finite rank and every chief factor in G is finite.

On PC-hypercentral and CC-hypercentral groups

This paper investigates the effects of the imposition of some chain conditions on groups having a generalized central series. It also provides a characterization of PC-groups and CC-groups with finite abelian section rank

On the Hirsch-Plotkin radical of a factorized group

Let the group G = AB be the product of two subgroups A and B. A normal subgroup K of G is said to be factorized if K = (A ∩ K)(B ∩ K) and A ∩ B ≤ K, and this is well-known to be equivalent to the fact that K = AK ∩ BK (see [1]). Easy examples show that normal subgroups of a product of two groups need not, in general, be factorized. Therefore the determination of certain special factorized subgroups is of relevant interest in the investigation concerning the structure of a factorized group. In this direction E. Pennington [5] proved that the Fitting subgroup of a finite product of two nilpotent groups is factorized. This result was extended to infinite groups by B. Amberg and theauthors, who provedin [2] that if the soluble group G = AB with finite abelian section rank isthe product of two locally nilpotent subgroups A and B, then the Hirsch-Plotkin radical (i.e. the maximum locally nilpotent normal subgroup) of G is factorized. If G is a soluble ℒI group and the factors A and B are nilpotent, it was shown in [3] that also the Fitting subgroup of G is factorized. However, Pennington's theorem becomes false for finite soluble groups which are the productof two arbitrary subgroups. For instance, the symmetric group of degree 4 is the product of a subgroup isomorphic with the symmetric group of degree 3 and a cyclic subgroup of order 4, but its Fitting subgroup is not factorized.

On trifactorized soluble groups of finite rank

Let the soluble-by-finite group G=AB=AC=BC be the product of two nilpotent subgroups A and B and a subgroup C. It is shown that, if G has finite abelian section rank and C is hypercentral (hypercyclic), then G is hypercentral (hypercyclic). Moreover, if G is an L 1-group and C is nilpotent, then G is nilpotent.

Groups with a nilpotent triple factorisation

In the investigation of factorised groups one often encounters groups G = AB = AK = BK which have a triple factorisation as a product of two subgroups A and B and a normal subgroup K of G. It is of particular interest to know whether G satisfies some nilpotency requirement whenever the three subgroups A, B and K satisfy this same nilpotency requirement. A positive answer to this problem for the classes of nilpotent, hypercentral and locally nilpotent groups is given under the hypothesis that K is a minimax group or G has finite abelian section rank. The results become false if K has only finite Prüfer rank. Some applications of the main theorems are pointed out.

Conjugacy separability and separable orbits

Two elements (subgroups) of a group G are called conjugacy separable if they are conjugate in G if and only if their images are conjugate in every finite quotient of G. The whole group is termed conjugacy separable if each pair of its elerr,ents is conjugacy separable. Conjugacy separable groups form a rather complicated class of groups. It is closed with respect to forming free products but not under taking subgroups, forming extensions and wreath products [S]. Restriction to the class of soluble groups does not help either. The best known result, the Theorem of Formanek [l] and Remeslennikov [7] yields that this class contains all polycyclic-by-finite groups. These groups also have all subgroups conjugacy separable (Grunewald, Segal [2]). In this paper we will prove conjugacy separability of elements and subgroups for a class of not necessarily finitely generated nilpotent groups of finite abelian section rank. Some examples will show that there is nearly no general way to extend these results to a wider class of groups. Regarding conjugacy as an operation of a group on itself yields another way how to extend theorems on conjugacy separability, which I found in a recent paper of Hilton and Roitberg [3]. Let Q be a group operating on a further group G. Two elements (subgroups) of G are said to bl: Q-conjugate, if there exists an element qE Q mapping the first element (subgr/Jup) onto the second. They are termed Q-separable if they are Q-conjugate or if there e!xists a finite Q-quotient of G in which their images are net Q-conjugate. G has separable Q-orbits if each pair of its elements is Q-separable. As usual Q is said to act nh’potently on G if it acts identically on the factors of a finite Q-invariant series of G and almost nilpotently if it conta:lns a subgroup of finite index acting nilpoltently on a Q-invariant subgroup of finite index of G. Hilton and Roitberg proved orbit separability for finitely generated nilpotent groups on which a finitely generated nilpotent group acts nilpotently. We will extend

Profinitely closed subgroups of soluble groups of finite rank

In his famous paper [6] Malcev proved that all subgroups of a soluble group of finite abelian section rank (as usual, we denote this class by -ip,) with no quasicyclic section are closed with respect to the profinite topology. He also showed that any torsion-free soluble group all subgroups of which are closed belongs to 9, and has no quasicyclic section. But it also seems to be of interest to obtain some information about (profinitely) closed subgroups of Y,-groups possessing quasicyclic sections (e.g., [ 1, Key Lemma]). In this paper we will characterize closed subgroups and investigate the relation between arbitrary subgroups and their closure in a class related to YO. Furthermore, we look at products and homomorphic images of closed subgroups of locally nilpotent groups. This work has mainly been motivated by the need of some results on (prolinitely) closed subgroups in order to prove theorems on conjugacy separability and on groups with the same finite images. But these results are of interest on their own and should therefore be written down together. Before stating our results we need some notation: The spectrum n(G) of a group G is the set of all primes p for which G has a quasicyclic p-section. If X is any class of groups we denote by S= its subclass of all reduced (i.e., with no radicable subgroups) S-groups of finite abelian section rank whose spectrum is contained in rr. If 7c is the set of all primes, we write Sr instead of Xx. We mainly use this definition in the case where 37 denotes the class d, JtT, JCJV and 9’ of all abelian, nilpotent, locally nilpotent and soluble groups, respectively. fl is the class of all finite groups. We put