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 Problems of teaching foreign students in mixed groups

2021 ◽  
Vol 127 ◽  
pp. 02012
Author(s):  
Elena Romualdovna Laskareva ◽  
Alina Aleksandrovna Pozdnyakova ◽  
Tatyana Nikolayevna Bazvanova ◽  
Daria Aleksandrovna Dmitrieva ◽  
Tatiana Pavlovna Chepkova
Keyword(s):  
General Education ◽  
Foreign Students ◽  
Negative Impact ◽  
Educational Institution ◽  
Educational Process ◽  
Mixed Group ◽  
Current Period ◽  
Group A ◽  
Foreign Speakers ◽  
Mixed Groups

The article analyzes the problems associated with teaching foreign students in mixed groups. A mixed group is a formally isolated part of the educational institution cohort that possesses such characteristics as: 1) a fixed heterogeneous ethno-cultural composition, 2) a different level of preparation of students for mastering the disciplines of the general educational program, 3) common cognitive interests and a single educational content in the current period of time; 4) joint educational activities under the guidance of the same teachers, 5) a single period of study. This article aims to highlight the range of language problems that hinder the development of general education programs by foreign students when studying as part of a mixed group. Since the structure of language difficulties varies for different categories of foreign citizens (foreigners with an initial level of proficiency in Russian, foreign speakers, bilinguals), it is important to have a differentiated approach to regulating the educational process and specially organized training taking into account the ethnic and linguistic needs of all members of the group. A differentiated approach to learning, building an individual way of each group member development allows mobilizing cognitive interests and motivating participants in the educational process. The development of specific recommendations to overcome the possible negative impact of a mixed group on the process of obtaining subject knowledge by foreign students is an important methodological task that requires further study.

Tensor Products and the Splitting of Abelian Groups

1971 ◽  
Vol 23 (5) ◽  
pp. 764-770 ◽  
Author(s):  
D. A. Lawver ◽  
E. H. Toubassi
Keyword(s):  
Positive Integer ◽  
Basic Assumption ◽  
Abelian Groups ◽  
Primary Component ◽  
Torsion Subgroup ◽  
Rank One ◽  
Splitting Problem ◽  
Torsion Subgroups ◽  
Free Element ◽  
Torsion Free

In [2], Irwin, Khabbaz, and Rayna discuss the splitting problem for abelian groups through the use of the tensor product. Throughout the paper they make a basic assumption, namely, that the torsion subgroup contains but one primary component. Under this restriction they introduce the concept of “splitting length”, which is a positive integer indicator of how far a group is from splitting. The results obtained along these lines may be extended to groups whose torsion subgroups contain any finite number of primary components by applying the work of Oppelt [4].Irwin, Khabbaz, and Rayna [2] define the notion of a p-sequence and show that for groups A where T(A) is p-primary and A/T(A) has rank one, the existence of a torsion-free element with a p-sequence is sufficient for the group to split.

On Direct Products of Abelian Groups

1970 ◽  
Vol 22 (3) ◽  
pp. 525-544 ◽  
Author(s):  
John M. Irwin ◽  
John D. O'Neill
Keyword(s):  
Finite Number ◽  
Abelian Groups ◽  
Torsion Subgroup ◽  
Direct Products ◽  
Index Set ◽  
Direct Sum ◽  
Primary Groups

In this paper we investigate the properties of the product (or complete direct sum) of torsion Abelian groups. The chief results concern products of Abelian primary groups (p-groups). Given a set of p-groups, [Gλ], over an index set Λ, the product of these groups is written λλ∈ΔGλ, the torsion subgroup of the product of these p-groups is written T[λGλ], and the discrete direct sum of the p-groups is written Σ Gλ.Definition. Σ Gλ is said to be an essentially bounded decomposition if and only if there exists an integer M > 0 such that MGλ = 0 for all but a finite number of Gλs; otherwise the decomposition is essentially unbounded.

scholarly journals Abelian group in a topos of sheaves: torsion and essential extensions

1989 ◽  
Vol 12 (1) ◽  
pp. 89-98
Author(s):  
Kiran R. Bhutani
Keyword(s):  
Abelian Group ◽  
Hausdorff Space ◽  
Torsion Group ◽  
Torsion Subgroup ◽  
Direct Sum ◽  
Group A ◽  
Torsion Groups ◽  
Essential Extensions

We investigate the properties of torsion groups and their essential extensions in the category AbShL of Abellan groups in a topos of sheaves on a locale. We show that every torsion group is a direct sum of its p-primary components and for a torsion group A, the group [A,B] is reduced for any BεAbShL.. We give an example to show that in AbShL the torsion subgroup of an injective group need not be injective. Further we prove that if the locale is Boolean or finite then essential extensions of torsion groups are torsion. Finally we show that for a first countable hausdorff space X essential extensions of torsion groups in AbSh0(X) are torsion iff X is discrete.

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 Algebraic entropies, Hopficity and co-Hopficity of direct sums of Abelian Groups

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Brendan Goldsmith ◽  
Ketao Gong
Keyword(s):  
Abelian Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Direct Sums ◽  
Direct Sum ◽  
Zero Entropy ◽  
Necessary And Sufficient

AbstractNecessary and sufficient conditions to ensure that the direct sum of two Abelian groups with zero entropy is again of zero entropy are still unknown; interestingly the same problem is also unresolved for direct sums of Hopfian and co-Hopfian groups.We obtain sufficient conditions in some situations by placing restrictions on the homomorphisms between the groups. There are clear similarities between the various cases but there is not a simple duality involved.

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 $A$.

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.

Sur les Sous-Groupes Purifiables des Groupes Abéliens Primaires

1990 ◽  
Vol 33 (1) ◽  
pp. 11-17 ◽  
Author(s):  
K. Benabdallah ◽  
C. Piché
Keyword(s):  
Abelian Groups ◽  
Divisible Group ◽  
Direct Sums ◽  
Cyclic Subgroups ◽  
Cyclic Groups ◽  
Direct Sum

AbstractThe class of primary abelian groups whose subsocles are purifiable is not yet completely characterized and it contains the class of direct sums of cyclic groups and torsion complete groups. In sharp constrast with this, the class of groups whose p2-bounded subgroups are purifiable consist only of those groups which are the direct sum of a bounded and a divisible group. Various tools are developed and a short application to the pure envelopes of cyclic subgroups is given in the last section.

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.

Sign in / Sign up

Export Citation Format

Share Document