scholarly journals Cyclically separated groups

1982 ◽  
Vol 26 (3) ◽  
pp. 355-384 ◽  
Author(s):  
Brian Hartley ◽  
John C. Lennox ◽  
Akbar H. Rhemtulla
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Finite Rank ◽  
Soluble Group ◽  
Residually Finite Groups ◽  
Interesting Class ◽  
Isolated Subgroup ◽  
Free Section ◽  
Torsion Free ◽  
Free Soluble Group

We call a group G cyclically separated if for any given cyclic subgroup B in G and subgroup A of finite index in B, there exists a normal subgroup N of G of finite index such that N ∩ B = A. This is equivalent to saying that for each element x ∈ G and integer n ≥ 1 dividing the order o(x) of x, there exists a normal subgroup N of G of finite index such that Nx has order n in G/N. As usual, if x has infinite order then all integers n ≥ 1 are considered to divide o(x). Cyclically separated groups, which are termed “potent groups” by some authors, form a natural subclass of residually finite groups and finite cyclically separated groups also form an interesting class whose structure we are able to describe reasonably well. Construction of finite soluble cyclically separated groups is given explicitly. In the discussion of infinite soluble cyclically separated groups we meet the interesting class of Fitting isolated groups, which is considered in some detail. A soluble group G of finite rank is Fitting isolated if, whenever H = K/L (L ⊲ K ≤ G) is a torsion-free section of G and F(H) is the Fitting subgroup of H then H/F(H) is torsion-free abelian. Every torsion-free soluble group of finite rank contains a Fitting isolated subgroup of finite index.

scholarly journals On groups with all subgroups almost subnormal

2004 ◽  
Vol 77 (2) ◽  
pp. 165-174 ◽  
Author(s):  
Eloisa Detomi
Keyword(s):  
Normal Subgroup ◽  
Nilpotent Group ◽  
Finite Index ◽  
Normal Closure ◽  
Fixed Integer ◽  
Torsion Free

AbstractIn this paper we consider groups in which every subgroup has finite index in the nth term of its normal closure series, for a fixed integer n. We prove that such a group is the extension of a finite normal subgroup by a nilpotent group, whose class is bounded in terms of n only, provided it is either periodic or torsion-free.

On Torsion-Free Groups of Finite Rank

1984 ◽  
Vol 36 (6) ◽  
pp. 1067-1080 ◽  
Author(s):  
David Meier ◽  
Akbar Rhemtulla
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Finite Rank ◽  
Maximal Element ◽  
Free Groups ◽  
Solvable Groups ◽  
Finitely Generated ◽  
Isolated Subgroup ◽  
Image Position ◽  
Torsion Free

This paper deals with two conditions which, when stated, appear similar, but when applied to finitely generated solvable groups have very different effect. We first establish the notation before stating these conditions and their implications. If H is a subgroup of a group G, let denote the setWe say G has the isolator property if is a subgroup for all H ≦ G. Groups possessing the isolator property were discussed in [2]. If we define the relation ∼ on the set of subgroups of a given group G by the rule H ∼ K if and only if , then ∼ is an equivalence relation and every equivalence class has a maximal element which may not be unique. If , we call H an isolated subgroup of G.

scholarly journals On the lattice of right ideals of the endomorphism ring of an abelian group

1988 ◽  
Vol 38 (2) ◽  
pp. 273-291 ◽  
Author(s):  
Theodore G. Faticoni
Keyword(s):  
Abelian Group ◽  
Finite Index ◽  
Endomorphism Ring ◽  
Finite Rank ◽  
Ideal Structure ◽  
Left Module ◽  
Linear Topology ◽  
Closed Submodules ◽  
The Right ◽  
Torsion Free

Let A be an abelian group, let ∧ = End (A), and assume that A is a flat left ∧-module. Then σ = { right ideals I ⊂ ∧ | IA = A} generates a linear topology oil ∧. We prove that Hom(A,·) is an equivalence from the category of those groups B ⊂ An satisfying B = Hom(A, B)A, onto the category of σ-closed submodules of finitely generated free right ∧-modules. Applications classify the right ideal structure of A, and classify torsion-free groups A of finite rank which are (nearly) isomorphic to each A-generated subgroup of finite index in A.

scholarly journals NORMAL AUTOMORPHISMS OF A FREE METABELIAN NILPOTENT GROUP

2009 ◽  
Vol 52 (1) ◽  
pp. 169-177
Author(s):  
GÉRARD ENDIMIONI
Keyword(s):  
Normal Subgroup ◽  
Nilpotent Group ◽  
Soluble Group ◽  
Groups Of Automorphisms ◽  
Inner Automorphisms ◽  
Free Soluble Group

AbstractAn automorphism φ of a group G is said to be normal if φ(H) = H for each normal subgroup H of G. These automorphisms form a group containing the group of inner automorphisms. When G is a non-abelian free (or free soluble) group, it is known that these groups of automorphisms coincide, but this is not always true when G is a free metabelian nilpotent group. The aim of this paper is to determine the group of normal automorphisms in this last case.

Groups in which each subnormal subgroup is commensurable with some normal subgroup

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ulderico Dardano ◽  
Fausto De Mari
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Soluble Group ◽  
Study Groups ◽  
Subnormal Subgroup

Abstract We study groups in which each subnormal subgroup is commensurable with a normal subgroup. Recall that two subgroups 𝐻 and 𝐾 are termed commensurable if H ∩ K H\cap K has finite index in both 𝐻 and 𝐾. Among other results, we show that if a (sub)soluble group 𝐺 has the above property, then 𝐺 is finite-by-metabelian, i.e., G ′′ G^{\prime\prime} is finite.

A Finite Index Property of Certain Solvable Groups

1984 ◽  
Vol 27 (4) ◽  
pp. 485-489
Author(s):  
A. H. Rhemtulla ◽  
H. Smith
Keyword(s):  
Solvable Group ◽  
Finite Index ◽  
Finite Rank ◽  
Solvable Groups ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Free Solvable Group ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent ◽  
Torsion Free

AbstractA group G is said to have the FINITE INDEX property (G is an FI-group) if, whenever H≤G, xp ∈ H for some x in G and p > 0, then |〈H, x〉: H| is finite. Following a brief discussion of some locally nilpotent groups with this property, it is shown that torsion-free solvable groups of finite rank which have the isolator property are FI-groups. It is deduced from this that a finitely generated torsion-free solvable group has an FI-subgroup of finite index if and only if it has finite rank.

scholarly journals Regulating hulls of almost completely decomposable groups

1993 ◽  
Vol 54 (2) ◽  
pp. 143-155 ◽  
Author(s):  
A. Mader ◽  
C. Vinsonhaler
Keyword(s):  
Finite Index ◽  
Finite Rank ◽  
Abelian Groups ◽  
Additive Group ◽  
Rational Numbers ◽  
Direct Sum ◽  
Free Abelian Groups ◽  
Decomposable Groups ◽  
The Relationship ◽  
Torsion Free

AbstractThis note investigates torsion-free abelian groups G of finite rank which embed, as subgroups of finite index, in a finite direct sum C of subgroups of the additive group of rational numbers. Specifically, we examine the relationship between G and C when the index of G in C is minimal. Some properties of Warfield duality are developed and used (in the case that G is locally free) to relate our results to earlier ones by Burkhardt and Lady.

scholarly journals COHOMOLOGY AND PROFINITE TOPOLOGIES FOR SOLVABLE GROUPS OF FINITE RANK

2012 ◽  
Vol 86 (2) ◽  
pp. 254-265 ◽  
Author(s):  
KARL LORENSEN
Keyword(s):  
Normal Subgroup ◽  
Solvable Group ◽  
Finite Index ◽  
Finite Rank ◽  
Solvable Groups ◽  
Graphic Module

AbstractAssume that G is a solvable group whose elementary abelian sections are all finite. Suppose, further, that p is a prime such that G fails to contain any subgroups isomorphic to Cp∞. We show that if G is nilpotent, then the pro-p completion map $G\to \hat {G}_p$ induces an isomorphism $H^\ast (\hat {G}_p,M)\to H^\ast (G,M)$ for any discrete $\hat {G}_p$-module M of finite p-power order. For the general case, we prove that G contains a normal subgroup N of finite index such that the map $H^\ast (\hat {N}_p,M)\to H^\ast (N,M)$ is an isomorphism for any discrete $\hat {N}_p$-module M of finite p-power order. Moreover, if G lacks any Cp∞-sections, the subgroup N enjoys some additional special properties with respect to its pro-p topology.

scholarly journals Representations of infinite soluble groups

1983 ◽  
Vol 24 (1) ◽  
pp. 43-52 ◽  
Author(s):  
Ian M. Musson
Keyword(s):  
Group Algebra ◽  
Soluble Group ◽  
Polycyclic Group ◽  
Soluble Groups ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Irreducible Modules ◽  
Torsion Free ◽  
Free Soluble Group

The purpose of this paper is to study the following two questions.(1) When does the group algebra of a soluble group have infinite dimensional irreducible modules?(2) When is the group algebra of a torsion free soluble group primitive?In relation to the first question, Roseblade [13] has proved that if G is a polycyclic group and k an absolute field then all irreducible kG-modules are finite dimensional. Here we prove a converse.

On the Cohomological Dimension of Soluble Groups

1981 ◽  
Vol 24 (4) ◽  
pp. 385-392 ◽  
Author(s):  
D. Gildenhuys ◽  
R. Strebel
Keyword(s):  
Homomorphic Image ◽  
Soluble Group ◽  
Cohomological Dimension ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Image Position ◽  
Torsion Free ◽  
Free Soluble Group

AbstractIt is known that every torsion-free soluble group G of finite Hirsch number hG is countable, and its homological and cohomological dimensions over the integers and rationals satisfy the inequalitiesWe prove that G must be finitely generated if the equality hG = cdQG holds. Moreover, we show that if G is a countable soluble group of finite Hirsch number, but not necessarily torsion-free, and if hG = cdQG, then hḠ = cdQḠ for every homomorphic image Ḡ of G.

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