Abelian Groups with a Vanishing Homology Group

1969 ◽  
Vol 21 ◽  
pp. 406-409 ◽  
Author(s):  
James A. Schafer
Keyword(s):  
Homology Group ◽  
Abelian Groups ◽  
Additive Group ◽  
Rational Numbers ◽  
Complete Characterization ◽  
Integral Homology ◽  
Positive Dimension ◽  
Homology Groups

In this paper, we wish to characterize those abelian groups whose integral homology groups vanish in some positive dimension. We obtain a complete characterization provided the dimension in which the homology vanishes is odd; in fact, we prove that the only abelian groups which possess a vanishing homology group in an odd dimension are, up to isomorphism, subgroups of Qn, where Q denotes the additive group of rational numbers. The case of vanishing in an even dimension is much more complicated. We exhibit a class of groups whose homology vanishes in even dimensions and is otherwise very nice, namely the subgroups of Q/Z, and then show that unless we impose further restrictions, there exist abelian groups which possess the homology of subgroups of Q/Z without being isomorphic to a subgroup of Q/Z.

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.

The abelianization of the congruence IA-automorphism group of a free group

2007 ◽  
Vol 142 (2) ◽  
pp. 239-248 ◽  
Author(s):  
TAKAO SATOH
Keyword(s):  
Automorphism Group ◽  
Free Group ◽  
Quotient Group ◽  
Homology Group ◽  
Integral Homology ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Homology Groups ◽  
Inner Automorphism

AbstractWe consider the abelianizations of some normal subgroups of the automorphism group of a finitely generated free group. Let Fn be a free group of rank n. For d ≥ 2, we consider a group consisting the automorphisms of Fn which act trivially on the first homology group of Fn with ${\mathbf Z}$/d${\mathbf Z}$-coefficients. We call it the congruence IA-automorphism group of level d and denote it by IAn,d. Let IOn,d be the quotient group of the congruence IA-automorphism group of level d by the inner automorphism group of a free group. We determine the abelianization of IAn,d and IOn,d for n ≥ 2 and d ≥ 2. Furthermore, for n=2 and odd prime p, we compute the integral homology groups of IA2,p for any dimension.

A Homological characterization of certain Abelian groups

1961 ◽  
Vol 57 (2) ◽  
pp. 256-264 ◽  
Author(s):  
A. J. Douglas
Keyword(s):  
Abelian Group ◽  
Homology Group ◽  
Abelian Groups ◽  
Unit Element ◽  
Homology Theory ◽  
Homology Groups ◽  
Complete Answer ◽  
Homological Characterization

Let G be a monoid; that is to say, G is a set such that with each pair σ, τ of elements of G there is associated a further element of G called the ‘product’ of σ and τ and written as στ. In addition it is required that multiplication be associative and that G shall have a unit element. The so-called ‘Homology Theory’† associates with each left G-module A and each integer n (n ≥ 0) an additive Abelian group Hn (G, A), called the nth homology group of G with coefficients in A. It is natural to ask what can be said about G if all the homology groups of G after the pth vanish identically in A. In this paper we give a complete answer to this question in the case when G is an Abelian group. Before describing the main result, however, it will be convenient to define what we shall call the homology type of G. We write Hn(G, A) ≡ 0 if Hn(G, A) = 0 for all left G-modules A.

scholarly journals One-relator groups and algebras related to polyhedral products

2021 ◽  
pp. 1-20
Author(s):  
Jelena Grbić ◽  
George Simmons ◽  
Marina Ilyasova ◽  
Taras Panov
Keyword(s):  
Homology Group ◽  
Homotopy Theory ◽  
Commutator Subgroup ◽  
Homology Groups ◽  
Homological Characterization ◽  
One Relator Groups ◽  
Combinatorial Condition ◽  
The Moment ◽  
Necessary And Sufficient

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$ , we specify a necessary and sufficient combinatorial condition for the commutator subgroup $RC_K'$ of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex $\mathcal {R}_K$ , to be a one-relator group; and for the Pontryagin algebra $H_{*}(\Omega \mathcal {Z}_K)$ of the moment-angle complex to be a one-relator algebra. We also give a homological characterization of these properties. For $RC_K'$ , it is given by a condition on the homology group $H_2(\mathcal {R}_K)$ , whereas for $H_{*}(\Omega \mathcal {Z}_K)$ it is stated in terms of the bigrading of the homology groups of $\mathcal {Z}_K$ .

scholarly journals Existence of Taut Foliations on Seifert Fibered Homology 3-spheres

2014 ◽  
Vol 66 (1) ◽  
pp. 141-169
Author(s):  
Shanti Caillat-Gibert ◽  
Daniel Matignon
Keyword(s):  
Homology Group ◽  
Integral Homology

AbstractThis paper concerns the problem of existence of taut foliations among 3-manifolds. From the work of David Gabai we know that a closed 3-manifold with non-trivial second homology group admits a taut foliation. The essential part of this paper focuses on Seifert fibered homology 3-spheres. The result is quite different if they are integral or rational but non-integral homology 3-spheres. Concerning integral homology 3-spheres, we can see that all but the 3-sphere and the Poincaré 3-sphere admit a taut foliation. Concerning non-integral homology 3-spheres, we prove there are infinitely many that admit a taut foliation, and infinitely many without a taut foliation. Moreover, we show that the geometries do not determine the existence of taut foliations on non-integral Seifert fibered homology 3-spheres.

On the Square Subgroup of a Mixed SI-Group

2018 ◽  
Vol 61 (1) ◽  
pp. 295-304 ◽  
Author(s):  
R. R. Andruszkiewicz ◽  
M. Woronowicz
Keyword(s):  
Abelian Group ◽  
Quotient Group ◽  
Abelian Groups ◽  
Additive Group ◽  
Negative Solution ◽  
New Class ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free ◽  
Group Modulo

AbstractThe relation between the structure of a ring and the structure of its additive group is studied in the context of some recent results in additive groups of mixed rings. Namely, the notion of the square subgroup of an abelian group, which is a generalization of the concept of nil-group, is considered mainly for mixed non-splitting abelian groups which are the additive groups only of rings whose all subrings are ideals. A non-trivial construction of such a group of finite torsion-free rank no less than two, for which the quotient group modulo the square subgroup is not a nil-group, is given. In particular, a new class of abelian group for which an old problem posed by Stratton and Webb has a negative solution, is indicated. A new, far from obvious, application of rings in which the relation of being an ideal is transitive, is obtained.

RATIONAL RINGS RELATED TO WEAKLY TRANSITIVE TORSION-FREE ABELIAN GROUPS

2009 ◽  
Vol 08 (05) ◽  
pp. 723-732 ◽  
Author(s):  
CHRIS MEEHAN ◽  
LUTZ STRÜNGMANN
Keyword(s):  
Abelian Groups ◽  
Rational Numbers ◽  
Rational Group ◽  
Free Abelian Groups ◽  
Torsion Free

We study subgroups R of the rational numbers ℚ having the property that for every pair of integers m, n such that gcd(m, n) = 1 and gcd(m, p) = gcd(n, p) = 1 whenever p is in the spectrum of R there is a unit u of R and an element r ∈ R such that un + rm = 1. These rings are closely related to weakly transitive separable groups. We prove that the property is dependent on the spectrum of the rational group in question and that the spectrum may be very complicated.

scholarly journals Homology group on manifolds and their foldings

2010 ◽  
Vol 41 (1) ◽  
pp. 31-38 ◽  
Author(s):  
M. Abu-Saleem
Keyword(s):  
Homology Group ◽  
Homology Groups ◽  
Definition Of

In this paper, we introduce the definition of the induced unfolding on the homology group. Some types of conditional foldings restricted on the elements of the homology groups are deduced. The effect of retraction on the homology group of a manifold is dicussed. The unfolding of variation curvature of manifolds on their homology group are represented. The relations between homology group of the manifold and its folding are deduced.

scholarly journals Zero square near-rings

1991 ◽  
Vol 51 (3) ◽  
pp. 497-504
Author(s):  
Patricia Jones
Keyword(s):  
Abelian Groups ◽  
Additive Group ◽  
Finite Abelian Groups

AbstractThe purpose of this paper is to provide examples and explore properties of a wide variety of zero square (left) near rings. Among the main results are complete classifications of (i) finite Abelian groups which are the additive group of a zero square near-ring and (ii) finite non-Abelian groups which support 3-nilpotent distributive zero square near-rings.

scholarly journals Antiautomorphisms and Biantiautomorphisms of Some Finite Abelian Groups

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 294
Author(s):  
Daniel López-Aguayo ◽  
Servando López Aguayo
Keyword(s):  
Lower Bound ◽  
Abelian Groups ◽  
Additive Group ◽  
Exact Formula ◽  
Finite Abelian Groups ◽  
Cyclic Groups ◽  
Partial Classification ◽  
Open Questions ◽  
Odd Order

We extend the concepts of antimorphism and antiautomorphism of the additive group of integers modulo n, given by Gaitanas Konstantinos, to abelian groups. We give a lower bound for the number of antiautomorphisms of cyclic groups of odd order and give an exact formula for the number of linear antiautomorphisms of cyclic groups of odd order. Finally, we give a partial classification of the finite abelian groups which admit antiautomorphisms and state some open questions.

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