scholarly journals On uniformly fully inert subgroups of abelian groups

2020 ◽  
Vol 8 (1) ◽  
pp. 5-27 ◽  
Author(s):  
Ulderico Dardano ◽  
Dikran Dikranjan ◽  
Luigi Salce
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Finite Index ◽  
Abelian Groups ◽  
Invariant Subgroup ◽  
Positive Answer ◽  
Natural Question ◽  
Direct Sum ◽  
Different Types ◽  
Dual Notion

AbstractIf H is a subgroup of an abelian group G and φ ∈ End(G), H is called φ-inert (and φ is H-inertial) if φ(H) ∩ H has finite index in the image φ(H). The notion of φ-inert subgroup arose and was investigated in a relevant way in the study of the so called intrinsic entropy of an endomorphism φ, while inertial endo-morphisms (these are endomorphisms that are H-inertial for every subgroup H) were intensively studied by Rinauro and the first named author.A subgroup H of an abelian group G is said to be fully inert if it is φ-inert for every φ ∈ End(G). This property, inspired by the “dual” notion of inertial endomorphism, has been deeply investigated for many different types of groups G. It has been proved that in some cases all fully inert subgroups of an abelian group G are commensurable with a fully invariant subgroup of G (e.g., when G is free or a direct sum of cyclic p-groups). One can strengthen the notion of fully inert subgroup by defining H to be uniformly fully inert if there exists a positive integer n such that |(H + φH)/H| ≤ n for every φ ∈ End(G). The aim of this paper is to study the uniformly fully inert subgroups of abelian groups. A natural question arising in this investigation is whether such a subgroup is commensurable with a fully invariant subgroup. This paper provides a positive answer to this question for groups belonging to several classes of abelian groups.

Criteria for Total Projectivity

1981 ◽  
Vol 33 (4) ◽  
pp. 817-825 ◽  
Author(s):  
Paul Hill
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Abelian Groups ◽  
General Criterion ◽  
Direct Sums ◽  
Primary Abelian Group ◽  
Cyclic Groups ◽  
Direct Sum ◽  
Totally Projective Groups ◽  
Torsion Groups

All groups herein are assumed to be abelian. It was not until the 1940's that it was known that a subgroup of an infinite direct sum of finite cyclic groups is again a direct sum of cyclics. This result rests on a general criterion due to Kulikov [7] for a primary abelian group to be a direct sum of cyclic groups. If G is p-primary, Kulikov's criterion presupposes that G has no elements (other than zero) having infinite p-height. For such a group G, the criterion is simply that G be the union of an ascending sequence of subgroups Hn where the heights of the elements of Hn computed in G are bounded by some positive integer λ(n). The theory of abelian groups has now developed to the point that totally projective groups currently play much the same role, at least in the theory of torsion groups, that direct sums of cyclic groups and countable groups played in combination prior to the discovery of totally projective groups and their structure beginning with a paper by R. Nunke [11] in 1967.

EXTREMAL CAYLEY DIGRAPHS OF FINITE ABELIAN GROUPS

2011 ◽  
Vol 12 (01n02) ◽  
pp. 125-135 ◽  
Author(s):  
ABBY GAIL MASK ◽  
JONI SCHNEIDER ◽  
XINGDE JIA
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Communication Networks ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Finite Abelian Groups ◽  
Element Subset ◽  
Cayley Digraph ◽  
Positive Integers ◽  
Element Set

Cayley digraphs of finite abelian groups are often used to model communication networks. Because of their applications, extremal Cayley digraphs have been studied extensively in recent years. Given any positive integers d and k. Let m*(d, k) denote the largest positive integer m such that there exists an m-element finite abelian group Γ and a k-element subset A of Γ such that diam ( Cay (Γ, A)) ≤ d, where diam ( Cay (Γ, A)) denotes the diameter of the Cayley digraph Cay (Γ, A) of Γ generated by A. Similarly, let m(d, k) denote the largest positive integer m such that there exists a k-element set A of integers with diam (ℤm, A)) ≤ d. In this paper, we prove, among other results, that [Formula: see text] for all d ≥ 1 and k ≥ 1. This means that the finite abelian group whose Cayley digraph is optimal with respect to its diameter and degree can be a cyclic group.

scholarly journals Type radicals and quasi-decompositions of torsion-free abelian groups

1992 ◽  
Vol 52 (2) ◽  
pp. 219-236 ◽  
Author(s):  
Oteo Mutzbauer
Keyword(s):  
Abelian Group ◽  
Finite Rank ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Torsion Free Abelian Group ◽  
Direct Sum ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractA composition sequence for a torsion-free abelian group A is an increasing sequenceof pure subgroups with rank 1 quotients and union A. Properties of A can be described by the sequence of types of these quotients. For example, if A is uniform, that is all the types in some sequence are equal, then A is complete decomposable if it is homogeneous. If A has finite rank and all permutations ofone of its type sequences can be realized, then A is quasi-isomorphic to a direct sum of uniform groups.

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 Extension of partial endomorphisms of abelian groups

1963 ◽  
Vol 6 (1) ◽  
pp. 45-48 ◽  
Author(s):  
C. G. Chehata
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Abelian Groups ◽  
Normal Subgroups ◽  
Extension Group ◽  
Image Position ◽  
Necessary And Sufficient

It is known [1] that for a partial endomorphism μ of a group G that maps the subgroup A ⊆ G onto B ⊆ G. G to be extendable to a total endomorphism μ* of a supergroup G* ⊆ G such that μ an isomorphism on G*(μ*)m for some positive integer m, it is necessary and sufficient that there exist in G a sequence of normal subgroupssuch that L1 ƞA is the kernel of μ andfor ι = 1, 2,…, m–1.The question then arises whether these conditions could be simplified when the group G is abelian. In this paper it is shown not only that the conditions are simplified when Gis abelian but also that the extension group G*⊇G can be chosen as an abelian group.

On Finite Essential Extensions of Torsion Free Abelian Groups

1996 ◽  
Vol 48 (5) ◽  
pp. 918-929 ◽  
Author(s):  
K. Benabdallah ◽  
M. A. Ouldbeddi
Keyword(s):  
Abelian Group ◽  
Finite Index ◽  
Free Abelian Group ◽  
Torsion Free Abelian Group ◽  
Direct Sum ◽  
Rank One ◽  
Free Abelian Groups ◽  
Decomposable Groups ◽  
Essential Extensions ◽  
Torsion Free

AbstractLet A be a torsion free abelian group. We say that a group K is a finite essential extension of A if K contains an essential subgroup of finite index which is isomorphic to A. Such K admits a representation as (A ℤ xkx)/ℤky where y = Nx + a for some k x k matrix N over Z and α ∈ Ak satisfying certain conditions of relative primeness and ℤk = {(α1,..., αk) : αi, ∈ ℤ}. The concept of absolute width of an f.e.e. K of A is defined and it is shown to be strictly smaller than the rank of A. A kind of basis substitution with respect to Smith diagonal matrices is shown to hold for homogeneous completely decomposable groups. This result together with general properties of our representations are used to provide a self contained proof that acd groups with two critical types are direct sum of groups of rank one and two.

A note on rank and direct decompositions of torsion-free Abelian groups. II

1969 ◽  
Vol 66 (2) ◽  
pp. 239-240 ◽  
Author(s):  
A. L. S. Corner
Keyword(s):  
Abelian Group ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Direct Sums ◽  
Torsion Free Abelian Group ◽  
Direct Sum ◽  
Free Abelian Groups ◽  
Rank 2 ◽  
Direct Decompositions ◽  
Torsion Free

According to well-known theorems of Kaplansky and Baer–Kulikov–Kapla nsky–Fuchs (4, 2), the class of direct sums of countable Abelian groups and the class of direct sums of torsion-free Abelian groups of rank 1 are both closed under the formation of direct summands. In this note I give an example to show that the class of direct sums of torsion-free Abelian groups of finite rank does not share this closure property: more precisely, there exists a torsion-free Abelian group G which can be written both as a direct sum G = A⊕B of 2 indecomposable groups A, B of rank ℵ0 and as a direct sum G = ⊕n ε zCn of ℵ0 indecomposable groups Cn (nεZ) of rank 2, where Z is the set of all integers.

scholarly journals On decomposable pseudofree groups

1999 ◽  
Vol 22 (3) ◽  
pp. 617-628
Author(s):  
Dirk Scevenels
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
The Other ◽  
Direct Sum ◽  
Other Hand ◽  
Rank One ◽  
Free Abelian Groups

An Abelian group is pseudofree of rankℓif it belongs to the extended genus ofℤℓ, i.e., its localization at every primepis isomorphic toℤpℓ. A pseudofree group can be studied through a sequence of rational matrices, the so-called sequential representation. Here, we use these sequential representations to study the relation between the product of extended genera of free Abelian groups and the extended genus of their direct sum. In particular, using sequential representations, we give a new proof of a result by Baer, stating that two direct sum decompositions into rank one groups of a completely decomposable pseudofree Abelian group are necessarily equivalent. On the other hand, sequential representations can also be used to exhibit examples of pseudofree groups having nonequivalent direct sum decompositions into indecomposable groups. However, since this cannot occur when using the notion of near-isomorphism rather than isomorphism, we conclude our work by giving a characterization of near-isomorphism for pseudofree groups in terms of their sequential representations.

Relations between Finite Homology and Homotopy

1969 ◽  
Vol 12 (2) ◽  
pp. 139-150
Author(s):  
B. Brown
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Finite Order ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Finitely Generated ◽  
Simply Connected

For a finite abelian group G let λ(G) be the least positive integer such that λ(G)G = 0. Let be the least integer such that λ(G) | (λ(G) divides ) and if 2 | λ(G) then 4 | . For a finitely generated abelian group G let GT be the subgroup of G made up of all elements of G of finite order, and let GF = G/GT. For a simply-connected C-W complex X, let be the smallest class of abelian groups containing the groups .

scholarly journals Rainbow connectivity of the non-commuting graph of a finite group

2016 ◽  
Vol 15 (07) ◽  
pp. 1650127 ◽  
Author(s):  
Yulong Wei ◽  
Xuanlong Ma ◽  
Kaishun Wang
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Positive Integer ◽  
Abelian Groups ◽  
Connection Number ◽  
Commuting Graph ◽  
Rainbow Connection Number ◽  
Rainbow Connection ◽  
Vertex Set ◽  
A Finite Group

Let [Formula: see text] be a finite non-abelian group. The non-commuting graph [Formula: see text] of [Formula: see text] has the vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text], where [Formula: see text] is the center of [Formula: see text]. We prove that the rainbow [Formula: see text]-connectivity of [Formula: see text] is [Formula: see text]. In particular, the rainbow connection number of [Formula: see text] is [Formula: see text]. Moreover, for any positive integer [Formula: see text], we prove that there exist infinitely many non-abelian groups [Formula: see text] such that the rainbow [Formula: see text]-connectivity of [Formula: see text] is [Formula: see text].

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