Set Theory Generated by Abelian Group Theory

1997 ◽  
Vol 3 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Paul C. Eklof
Keyword(s):  
Group Theory ◽  
Abelian Group ◽  
Set Theory ◽  
Identity Element ◽  
Historical Context ◽  
Group Operation ◽  
Independence Result ◽  
Comprehensive Survey ◽  
New Ideas ◽  
Free Abelian Groups

Introduction. This survey is intended to introduce to logicians some notions, methods and theorems in set theory which arose—largely through the work of Saharon Shelah—out of (successful) attempts to solve problems in abelian group theory, principally the Whitehead problem and the closely related problem of the existence of almost free abelian groups. While Shelah's first independence result regarding the Whitehead problem used established set-theoretical methods (discussed below), his later work required new ideas; it is on these that we focus. We emphasize the nature of the new ideas and the historical context in which they arose, and we do not attempt to give precise technical definitions in all cases, nor to include a comprehensive survey of the algebraic results.In fact, very little algebraic background is needed beyond the definitions of group and group homomorphism. Unless otherwise specified, we will use the word “group” to refer to an abelian group, that is, the group operation is commutative. The group operation will be denoted by +, the identity element by 0, and the inverse of a by −a. We shall use na as an abbreviation for a + a + … + a [n times] if n is positive, and na = (−n)(−a) if n is negative.

On the square subgroups of decomposable torsion-free abelian groups of rank three

2016 ◽  
Vol 7 (4) ◽  
Author(s):  
Fysal Hasani ◽  
Fatemeh Karimi ◽  
Alireza Najafizadeh ◽  
Yousef Sadeghi
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractThe square subgroup of an abelian group

On the SL(2, C)-representation rings of free abelian groups

2018 ◽  
Vol 167 (02) ◽  
pp. 229-247
Author(s):  
TAKAO SATOH
Keyword(s):  
Abelian Group ◽  
Upper Bound ◽  
Automorphism Groups ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Free Groups ◽  
The Other ◽  
Subsequent Research ◽  
Representation Rings ◽  
Free Abelian Groups

AbstractIn this paper, we study “the ring of component functions” of SL(2, C)-representations of free abelian groups. This is a subsequent research of our previous work [11] for free groups. We introduce some descending filtration of the ring, and determine the structure of its graded quotients.Then we give two applications. In [30], we constructed the generalized Johnson homomorphisms. We give an upper bound on their images with the graded quotients. The other application is to construct a certain crossed homomorphisms of the automorphism groups of free groups. We show that our crossed homomorphism induces Morita's 1-cocycle defined in [22]. In other words, we give another construction of Morita's 1-cocyle with the SL(2, C)-representations of the free abelian group.

Skew-morphisms of cyclic 2-groups

2019 ◽  
Vol 22 (4) ◽  
pp. 617-635
Author(s):  
Shaofei Du ◽  
Kan Hu
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Identity Element ◽  
A Finite Group

AbstractA skew-morphism of a finite group A is a permutation φ on A fixing the identity element, and for which there exists an integer function π on A such that, for all {x,y\in A}, {\varphi(xy)=\varphi(x)\varphi^{\pi(x)}(y)}. In [I. Kovács and R. Nedela, Skew-morphisms of cyclic p-groups, J. Group Theory 20 2017, 6, 1135–1154], Kovács and Nedela determined skew-morphisms of the cyclic p-groups for any odd prime p. In this paper, we shall determine that of cyclic 2-groups.

Cellular covers of ℵ1-free abelian groups

2015 ◽  
Vol 14 (10) ◽  
pp. 1550139 ◽  
Author(s):  
José L. Rodríguez ◽  
Lutz Strüngmann
Keyword(s):  
Abelian Group ◽  
Exact Sequence ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Countable Rank ◽  
Free Abelian Groups ◽  
Rank 2 ◽  
Exact Sequences ◽  
Torsion Free ◽  
Cellular Cover

In this paper, we first show that for every natural number n and every countable reduced cotorsion-free group K there is a short exact sequence [Formula: see text] such that the map G → H is a cellular cover over H and the rank of H is exactly n. In particular, the free abelian group of infinite countable rank is the kernel of a cellular exact sequence of co-rank 2 which answers an open problem from Rodríguez–Strüngmann [J. L. Rodríguez and L. Strüngmann, Mediterr. J. Math.6 (2010) 139–150]. Moreover, we give a new method to construct cellular exact sequences with prescribed torsion free kernels and cokernels. In particular we apply this method to the class of ℵ1-free abelian groups in order to complement results from the cited work and Göbel–Rodríguez–Strüngmann [R. Göbel, J. L. Rodríguez and L. Strüngmann, Fund. Math.217 (2012) 211–231].

Technical and Engineering Challenges Faced in Design of the Gelugor Mainline, Penang, Malaysia

2000 ◽  
Author(s):  
Ahmad Khairiri ◽  
Abdul Ghani
Keyword(s):  
Power Plant ◽  
Selection Process ◽  
Peninsular Malaysia ◽  
Gas Utilization ◽  
Gravity Flows ◽  
Comprehensive Survey ◽  
New Ideas ◽  
Environmentally Sensitive ◽  
Pipeline Design ◽  
Electrical Cables

PETRONAS Gas Berhad (PGB), a listed partly-owned subsidiary of PETRONAS is undertaking the Peninsular Gas Utilization Project - Stage III (PGU III). The PGU III Sector 2 & 3 project extends from Lumut, Perak to Pauh, Perlis on the West Coast of Peninsular Malaysia, with two laterals, the Gurun Lateral and the Penang Lateral. The Penang Lateral is further divided into the Prai Lateral, Gelugor Mainline and the Gelugor Tenaga National Berhad (TNB) Meter Station project individually. This paper will focus its’ discussion on the technical and engineering challenges faced in the design of the Gelugor Mainline Project. The Gelugor Mainline consist of about 7 km NPS 24 gas pipeline which runs from TNB Prai Power Plant of the mainline to TNB Gelugor Power Plant on the Penang Island. The pipeline is planned to operate at high pressure. The pipeline route traverses complex terrain with varied seabed lithology with depth up to 16m. The pipeline will cross active shipping lane, zones of live electrical cables, extensive system of submarine pipeline, mudbank and areas susceptible to mass gravity flows. There is also an area subjected to future development with an expected 170 kN/m2 distributed load with 3 m covers required which is an expressway and Light Rapid Transportation (LRT) system, the proposed development of a “Vision City” is also a few meters away from the pipeline. The pipeline also passes by an environmentally sensitive area that includes the rearing of caged fishing activities. It is essential for the project team to determine and investigate the variable risks to the pipeline in order to take appropriate evasive and mitigating action in design. This required comprehensive survey activities in the busy shipping straits. The paper will explore and detail out its’ study starting from the description of the design concept to the varied environmental challenges. This paper aims to describe the route selection process, the pipeline design process and outline some of the specialist techniques employed in the design construction of the Gelugor Mainline, Penang, Malaysia. In nearly every major project a drive for cost and risk reduction results in a number of new ideas with varying degree of impact. Hence, the following areas will be elaborated and presented in greater details: • Pipeline Design. • Pipe-lay vessel. • Pipeline installation methodology.

A Classification Of n-Abelian Groups

1969 ◽  
Vol 21 ◽  
pp. 1238-1244 ◽  
Author(s):  
J. L. Alperin
Keyword(s):  
Group Theory ◽  
Abelian Group ◽  
Recent Result ◽  
Abelian Groups ◽  
The Way

The concept of an abelian group is central to group theory. For that reason many generalizations have been considered and exploited. One, in particular, is the idea of an n-abelian group (6). If n is an integer and n > 1, then a group G is n-abelian if, and only if,(xy)n = xnynfor all elements x and y of G. Thus, a group is 2-abelian if, and only if, it is abelian, while non-abelian n-abelian groups do exist for every n > 2.Many results pertaining to the way in which groups can be constructed from abelian groups can be generalized to theorems on n-abelian groups (1; 2). Moreover, the case of n = p, a prime, is useful in the study of finite p-groups (3; 4; 5). Moreover, a recent result of Weichsel (9) gives a description of all p-abelian finite p-groups.

scholarly journals Almost-Free E-Rings of Cardinality ℵ1

2003 ◽  
Vol 55 (4) ◽  
pp. 750-765
Author(s):  
Rüdiger Göbel ◽  
Saharon Shelah ◽  
Lutz Strüngmann
Keyword(s):  
Abelian Group ◽  
Set Theory ◽  
Regular Cardinal ◽  
Additive Structure ◽  
Unital Ring ◽  
Ring Element ◽  
Models Of Set Theory

AbstractAn E-ring is a unital ring R such that every endomorphism of the underlying abelian group R+ is multiplication by some ring element. The existence of almost-free E-rings of cardinality greater than 2ℵ0 is undecidable in ZFC. While they exist in Gödel's universe, they do not exist in other models of set theory. For a regular cardinal ℵ1 ≤ λ 2ℵ0 we construct E-rings of cardinality λ in ZFC which have ℵ1-free additive structure. For λ = ℵ1 we therefore obtain the existence of almost-free E-rings of cardinality ℵ1 in ZFC.

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