scholarly journals ABELIAN GROUPS WITH A p2-BOUNDED SUBGROUP, REVISITED

2011 ◽  
Vol 10 (03) ◽  
pp. 377-389
Author(s):  
CARLA PETRORO ◽  
MARKUS SCHMIDMEIER
Keyword(s):  
Abelian Group ◽  
Prime Number ◽  
Maximal Ideal ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Uniserial Ring

Let Λ be a commutative local uniserial ring of length n, p be a generator of the maximal ideal, and k be the radical factor field. The pairs (B, A) where B is a finitely generated Λ-module and A ⊆B a submodule of B such that pmA = 0 form the objects in the category [Formula: see text]. We show that in case m = 2 the categories [Formula: see text] are in fact quite similar to each other: If also Δ is a commutative local uniserial ring of length n and with radical factor field k, then the categories [Formula: see text] and [Formula: see text] are equivalent for certain nilpotent categorical ideals [Formula: see text] and [Formula: see text]. As an application, we recover the known classification of all pairs (B, A) where B is a finitely generated abelian group and A ⊆ B a subgroup of B which is p2-bounded for a given prime number p.

A Spectral Sequence for Cohomotopy

1969 ◽  
Vol 21 ◽  
pp. 712-729 ◽  
Author(s):  
Benson Samuel Brown
Keyword(s):  
Abelian Group ◽  
Spectral Sequence ◽  
Prime Number ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
Image Position

For a prime number p let be the class of finite abelian groups whose orders are prime to p. For a finitely generated abelian group G, let Gp be the sum of the free and p-primary components of G. Our aim in this paper is to prove the following theorem.Theorem. Suppose that(i) Hi(X;Z) = 0 for i > k,(ii) for i > k – dThen there exists a spectral sequence withand the differential is given by

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 Homogeneity of inverse semigroups

2018 ◽  
Vol 28 (05) ◽  
pp. 837-875 ◽  
Author(s):  
Thomas Quinn-Gregson
Keyword(s):  
Inverse Semigroup ◽  
Finite Groups ◽  
Abelian Groups ◽  
Semigroup Theory ◽  
Inverse Semigroups ◽  
Finitely Generated ◽  
Homogeneous Groups ◽  
Inverse Subsemigroups

An inverse semigroup [Formula: see text] is a semigroup in which every element has a unique inverse in the sense of semigroup theory, that is, if [Formula: see text] then there exists a unique [Formula: see text] such that [Formula: see text] and [Formula: see text]. We say that a countable inverse semigroup [Formula: see text] is a homogeneous (inverse) semigroup if any isomorphism between finitely generated (inverse) subsemigroups of [Formula: see text] extends to an automorphism of [Formula: see text]. In this paper, we consider both these concepts of homogeneity for inverse semigroups, and show when they are equivalent. We also obtain certain classifications of homogeneous inverse semigroups, in particular periodic commutative inverse semigroups. Our results may be seen as extending both the classification of homogeneous semilattices and the classification of certain classes of homogeneous groups, such as homogeneous abelian groups and homogeneous finite groups.

Hypercyclic Abelian Groups of Affine Maps on ℂn

2013 ◽  
Vol 56 (3) ◽  
pp. 477-490 ◽  
Author(s):  
Adlene Ayadi
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Dense Orbit ◽  
Finitely Generated ◽  
Affine Maps

Abstract.We give a characterization of hypercyclic abelian group 𝒢 of affine maps on ℂn. If G is finitely generated, this characterization is explicit. We prove in particular that no abelian group generated by n affine maps on Cn has a dense orbit.

scholarly journals Commensurability and QI classification of free products of finitely generated abelian groups

2008 ◽  
Vol 137 (03) ◽  
pp. 811-813 ◽  
Author(s):  
Jason A. Behrstock ◽  
Tadeusz Januszkiewicz ◽  
Walter D. Neumann
Keyword(s):  
Abelian Groups ◽  
Finitely Generated ◽  
Free Products

Representations of holomorphs of group extensions with Abelian kernels

1975 ◽  
Vol 78 (3) ◽  
pp. 357-368 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Finite Rank ◽  
Automorphism Groups ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Soluble Groups ◽  
Group Extensions

This paper is devoted to the construction of faithful representations of the automorphism group and the holomorph of an extension of an abelian group by some other group, the representations here being homomorphisms into certain restricted parts of the automorphism groups of smallish abelian groups. We apply these to two very specific cases, namely to finitely generated metabelian groups and to certain soluble groups of finite rank. We describe the applications first.

scholarly journals Elementary equivalence for finitely generated nilpotent groups and multilinear maps

1998 ◽  
Vol 58 (3) ◽  
pp. 479-493 ◽  
Author(s):  
Francis Oger
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Elementary Equivalence ◽  
Linear Map ◽  
Multilinear Maps

We show that two finitely generated finite-by-nilpotent groups are elementarily equivalent if and only if they satisfy the same sentences with two alternations of quantifiers. For each integer n ≥ 2, we prove the same result for the following classes of structures:(1) the (n + 2)-tuples (A1, …, An+1, f), where A1, …, An+1 are disjoint finitely generated Abelian groups and f: A1 × … × An → An+1 is a n-linear map;(2) the triples (A, B, f), where A, B are disjoint finitely generated Abelian groups and f: An → B is a n-linear map;(3) the pairs (A, f), where A is a finitely generated Abelian group and f: An → A is a n-linear map.In the proof, we use some properties of commutative rings associated to multilinear maps.

scholarly journals Maximal perpendicularity in certain Abelian groups

2017 ◽  
Vol 9 (1) ◽  
pp. 235-247
Author(s):  
Mika Mattila ◽  
Jorma K. Merikoski ◽  
Pentti Haukkanen ◽  
Timo Tossavainen
Keyword(s):  
Abelian Group ◽  
Vector Space ◽  
Binary Relation ◽  
Abelian Groups ◽  
The Other ◽  
Finitely Generated ◽  
Other Hand

AbstractWe define perpendicularity in an Abelian group G as a binary relation satisfying certain five axioms. Such a relation is maximal if it is not a subrelation of any other perpendicularity in G. A motivation for the study is that the poset (𝒫, ⊆) of all perpendicularities in G is a lattice if G has a unique maximal perpendicularity, and only a meet-semilattice if not. We study the cardinality of the set of maximal perpendicularities and, on the other hand, conditions on the existence of a unique maximal perpendicularity in the following cases: G ≅ ℤn, G is finite, G is finitely generated, and G = ℤ ⊕ ℤ ⊕ ⋯. A few such conditions are found and a few conjectured. In studying ℝn, we encounter perpendicularity in a vector space.

scholarly journals Subshifts and colorings on ascending HNN-extensions of finitely generated abelian groups

2021 ◽  
pp. 1-26
Author(s):  
EDUARDO SILVA
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Global Constraints ◽  
Finitely Generated ◽  
Subshift Of Finite Type ◽  
Mixing Properties ◽  
Hnn Extensions ◽  
Hnn Extension ◽  
Periodic Configuration ◽  
The Right

Abstract For an ascending HNN-extension $G*_{\psi }$ of a finitely generated abelian group G, we study how a synchronization between the geometry of the group and weak periodicity of a configuration in $\mathcal {A}^{G*_{\psi }}$ forces global constraints on it, as well as in subshifts containing it. A particular case are Baumslag–Solitar groups $\mathrm {BS}(1,N)$ , $N\ge 2$ , for which our results imply that a $\mathrm {BS}(1,N)$ -subshift of finite type which contains a configuration with period $a^{N^\ell }\!, \ell \ge 0$ , must contain a strongly periodic configuration with monochromatic $\mathbb {Z}$ -sections. Then we study proper n-colorings, $n\ge 3$ , of the (right) Cayley graph of $\mathrm {BS}(1,N)$ , estimating the entropy of the associated subshift together with its mixing properties. We prove that $\mathrm {BS}(1,N)$ admits a frozen n-coloring if and only if $n=3$ . We finally suggest generalizations of the latter results to n-colorings of ascending HNN-extensions of finitely generated abelian groups.

An Approximation to {X, Y}

1969 ◽  
Vol 12 (2) ◽  
pp. 133-137
Author(s):  
B. Brown
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Cohomology Groups ◽  
Finitely Generated ◽  
Finite Abelian Groups

All homology and cohomology groups are reduced. is the class of finite abelian groups whose orders are prime to p. is the class of abelian groups whose orders are products of primes less than a. If G is a finitely generated abelian group, Gp is the quotient of G by the subgroup of G made up of all elements whose orders are prime to p.

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