scholarly journals Transformations on tensor products and the torsionless property in abelian groups: Corrigendum

1977 ◽  
Vol 16 (2) ◽  
pp. 319-319
Author(s):  
Cary Webb
Keyword(s):  
Abelian Groups ◽  
Tensor Products ◽  
Correct Formulation ◽  
Group A ◽  
Slender Group

Professor Paul L. Sperry, when reviewing the paper [1] for Mathematical Reviews, has drawn the author's attention to an error in the formulation of Theorem 4.The hypotheses on the abelian groups Ci, i ∈ I, in Theorem 4 were incorrectly stated. (They implied, for one thing, that the slender group A is divisible, an absurdity.) The correct formulation is as follows.

A generalization of co-Hopfian abelian groups

2017 ◽  
Vol 27 (04) ◽  
pp. 351-360
Author(s):  
Andrey Chekhlov ◽  
Peter Danchev
Keyword(s):  
Abelian Group ◽  
Representation Theorem ◽  
Abelian Groups ◽  
Classical Notion ◽  
Group A ◽  
Hopfian Group ◽  
Comprehensive Study

We define the concept of an [Formula: see text]-co-Hopfian abelian group, which is a nontrivial generalization of the classical notion of a co-Hopfian group. A systematic and comprehensive study of these groups is given in very different ways. Specifically, a representation theorem for [Formula: see text]-co-Hopfian groups is established as well as it is shown that there exists an [Formula: see text]-co-Hopfian group which is not co-Hopfian.

scholarly journals On solving systems of random linear disequations

2008 ◽  
Vol 8 (6&7) ◽  
pp. 579-594
Author(s):  
G. Ivanyos
Keyword(s):  
Abelian Groups ◽  
Main Idea ◽  
Constant Factor ◽  
Linear Functions ◽  
Important Special Case ◽  
Hidden Subgroup Problem ◽  
Left Hand ◽  
Shift Problem ◽  
Group A ◽  
Subgroup Problem

An important special case of the hidden subgroup problem is equivalent to the hidden shift problem over abelian groups. An efficient solution to the latter problem could serve as a building block of quantum hidden subgroup algorithms over solvable groups. The main idea of a promising approach to the hidden shift problem is a reduction to solving systems of certain random disequations in finite abelian groups. By a disequation we mean a constraint of the form $f(x)\neq 0$. In our case, the functions on the left hand side are generalizations of linear functions. The input is a random sample of functions according to a distribution which is up to a constant factor uniform over the "linear" functions $f$ such that $f(u)\neq 0$ for a fixed, although unknown element $u\in A$. The goal is to find $u$, or, more precisely, all the elements $u'\in A$ satisfying the same disequations as $u$. In this paper we give a classical probabilistic algorithm which solves the problem in an abelian $p$-group $A$ in time polynomial in the sample size $N$, where $N=(\log\size{A})^{O(q^2)}$, and $q$ is the exponent of $A$.

Construction of orders in Abelian groups

1961 ◽  
Vol 57 (3) ◽  
pp. 476-482 ◽  
Author(s):  
H.-H. Teh
Keyword(s):  
Abelian Group ◽  
Binary Relation ◽  
Abelian Groups ◽  
Group A

Let G be an Abelian group. A binary relation ≥ denned in G is called an order of G if for each x, y, z ε G,(i) x ≥ y or y ≥ x (and hence x ≥ x);(ii) x ≥ y and y ≥ x ⇒ x = y,(if x ≥ y and x ≠ y, we write x > y);(iii) x ≥ y and y ≥ z ⇒ x = z;(iv) z ≥ y ⇒ x + z ≥ y + z.

Mixed Abelian Groups

1967 ◽  
Vol 19 ◽  
pp. 1259-1262 ◽  
Author(s):  
John A. Oppelt
Keyword(s):  
Abelian Groups ◽  
Torsion Subgroup ◽  
Mixed Group ◽  
Direct Sum ◽  
Group A ◽  
Splitting Problem ◽  
Mixed Groups

The difficulties encountered in the theory of mixed Abelian groups can become decidedly less complex, if it is possible to reduce the question to mixed groups whose torsion subgroup is 𝒫-primary. Call such a group a p-mixed group. In §1 we show that the splitting problem for a mixed group is reducible to the same problem for certain associated 𝒫-mixed groups. In §2 we look at groups which are a direct sum of 𝒫-mixed groups.

scholarly journals On Injective Sheaves

1964 ◽  
Vol 7 (3) ◽  
pp. 415-423
Author(s):  
H. Kleisli ◽  
Y.C. Wu
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Crucial Point ◽  
Arbitrary Ring ◽  
Injective Modules ◽  
Group A ◽  
Group B ◽  
Divisible Groups ◽  
Group D ◽  
Divisible Abelian Group

A divisible abelian group D can be characterized by the following property: Every homomorphism from an abelian group A to D can be extended to every abelian group B containing A. This together with the result that every abelian group can be embedded in a divisible group is a crucial point in many investigations on abelian groups. It was Baer, [1], who extended this result to modules over an arbitrary ring, replacing divisible groups by injective modules, that is, modules with the property mentioned above. Another proof was found later by Eckmann and Schopf, [3]. This proof assumes the proposition to hold for abelian groups and transfers it in a very simple and elegant manner to modules. In the sequel, we shall refer to this proof as to the Eckmann-Schopf proof.

Tensor products of Abelian groups

1938 ◽  
Vol 4 (3) ◽  
pp. 495-528 ◽  
Author(s):  
Hassler Whitney
Keyword(s):  
Abelian Groups ◽  
Tensor Products

scholarly journals Galkin Quandles, Pointed Abelian Groups, and Sequence A000712

10.37236/2676 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
William E. Clark ◽  
Xiang-dong Hou
Keyword(s):  
Abelian Group ◽  
Algebraic Structure ◽  
Abelian Groups ◽  
Knot Invariants ◽  
Group A

For each pointed abelian group $(A,c)$, there is an associated Galkin quandle $G(A,c)$ which is an algebraic structure defined on $\Bbb Z_3\times A$ that can be used to construct knot invariants. It is known that two finite Galkin quandles are isomorphic if and only if their associated pointed abelian groups are isomorphic. In this paper we classify all finite pointed abelian groups. We show that the number of nonisomorphic pointed abelian groups of order $q^n$ ($q$ prime) is $\sum_{0\le m\le n}p(m)p(n-m)$, where $p(m)$ is the number of partitions of integer $m$.

scholarly journals Skew Tensor Products and Groups of Class Two

1963 ◽  
Vol 23 ◽  
pp. 15-51 ◽  
Author(s):  
Paul Conrad
Keyword(s):  
Nilpotent Group ◽  
Abelian Groups ◽  
Tensor Products ◽  
Extension Theory

Let Δ and N be abelian groups and let f be a mapping of Δ × Δ into N that is bilinear, skew symmetric and satisfies f(α, α) = 0 for all α ∈ Δ. Such a mapping f is called a (*)-mapping. By the Schreier extension theory Δ, N and f determine a nilpotent group G(Δ, N, f) of class two that consists of the set Δ × N with composition(α, a) + (β, b) = (α + β, f(α, β) + a + b).

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