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Friday, May 8, 2020 | History

3 edition of Some results on direct sums of Hopfian groups. found in the catalog.

Some results on direct sums of Hopfian groups.

Ronald Hirshon

# Some results on direct sums of Hopfian groups.

## by Ronald Hirshon

Published in [Garden City, N.Y.] .
Written in English

Subjects:
• Hopfian groups.

• Edition Notes

Classifications
LC ClassificationsQA171 .H66
The Physical Object
Paginationiv, 78 l.
Number of Pages78
ID Numbers
Open LibraryOL3903242M
LC Control Number81465699

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In mathematics, specifically in group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.. In the context of abelian groups, the direct product is sometimes referred to. famous text An Introduction to Probability Theory and Its Applications (New York: Wiley, ). In the preface, Feller wrote about his treatment of ﬂuctuation in coin tossing: “The results are so amazing and so at variance with common intuition that even sophisticated colleagues doubted that coins actually misbehave as theory by:

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### Some results on direct sums of Hopfian groups by Ronald Hirshon Download PDF EPUB FB2

Buy Some results on direct sums of Hopfian groups on FREE SHIPPING on qualified orders. finitely generated Hopfian groups is Hopfian-and it is easy to see that a free product is non-Hopfian if one of the factors is non-Hopfian.

Our contribution is the following theorem. Theorem. Let G = (ax, a2, t; t~xw't = wm) where w is a word in ax, a2 that is not primitive and not a proper power in the free group F(ax, a^. Then G is Hopfian. Some Classes of Hopfian Abelian Groups. Direct sums of cyclic groups that are Hopfian groups are characterized.

Finally, the paper looks at specific examples of rings whose additive groups are. Torsion-free $\delta$-hyperbolic groups are hopfian, and it's a theorem of Sela that one-ended torsion free hyperbolic groups are co-hopfian (Z. Sela. Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups.

Geom. Funct. Anal., 7(3)–, ). The main results proved in this note are the following:(i) Any finitely generated group can be expressed as a quotient of a finitely presented, centreless group which is simultaneously Hopfian and. Abstract: An algebra is said to be hopfian if it is not isomorphic to a proper quotient of itself.

We describe several classes of hopfian and of non-hopfian unital lattice-ordered abelian groups and : Daniele Mundici. This paper investigates the preservation of hopficity and co-hopficity on passing to finite-index subsemigroups and extensions. It was already known that hopficity is not preserved on passing to finite Rees index subsemigroups, even in the finitely generated case.

We give a stronger example to show that it is not preserved even in the finitely presented case. It was also known that hopficity Author: Alan J. Cain, Victor Maltcev. The notions of Hopfian and co-Hopfian groups have been of interest for some time. In this present work we exploit some unpublished ideas of Corner to answer questions relating to such groups.

Is there a simple group (all normal subgroups are the group itself or the trivial group) that is non co-hopfian (there exists an injective group endomorphism that is not surjective). abstract-algebra group-theory hopfian-groups. Can someone give me some applications of Hopfian and co- Stack Exchange Network Stack Exchange network consists of Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

The main result (Theorem ) states that the direct product of two closed orientable manifolds (of different dimension) homotopically determined by π 1 and with coperfectly Hopfian fundamental groups (one normally incommensurable with the other one) is a shape m simpl o-fibrator, if it is a Hopfian manifold and a codimension-2 : Violeta Vasilevska.

Request PDF | The Hopfian exponent of an abelian group | If $$G$$ is a Hopfian abelian group then it is, in general, difficult to determine if direct sums of copies of $$G$$ will remain Hopfian. We show that a soluble group satisfying the minimal condition for its normal subgroups is co-hopfian and that a torsion-free finitely generated soluble group of finite rank is hopfian.

The latter property is a consequence of a stronger result: in a minimax soluble group, the kernel of an endomorphism is finite if and only if its image is of finite index in the by: 1.

In this paper we give a survey of some recent results on hopficity, co-hopficity, residual finiteness etc., in many familiar categories, after briefly describing the origin of the : Daniele Mundici. MORE RESULTS ON COMMUTATOR SUBGROUPS. INVARIANT SERIES AND CHIEF SERIES.

KEY WORDS. SUMMARY. SELF ASSESMENT QUESTIONS. SUGGESTED READINGS. OBJECTIVE. Objective of this Chapter is to study some properties of groups by studying the properties of the series of its subgroups and factor groups. Size: KB. As Dietrich Burde has pointed out, you need to consider infinitely generated groups.

This is because of the following result due to Mal'cev. Theorem (Mal'cev): A finitely generated, residually finite group is Hopfian.

The purpose of these notes is to provide readers with some basic insight into group theory as quickly as possible. Prerequisites for this paper are direct sum or direct product of two groups Some practice for readers unfamiliar with formal logic notation: want to consult a book on axiomatic set Size: KB.

Basic Deﬁnitions and Results The axioms for a group are short and natural. Yet somehow hidden behind these axioms is the monster simple group, a huge and extraordinary mathematical object, which appears to rely on numerous bizarre coincidences to exist.

The axioms for groups give no obvious hint that anything like this Size: KB. In mathematics, a group G is called the direct sum of two subgroups H1 and H2 if each H1 and H2 are normal subgroups of subgroups H1.

Definition.A group is called hopfian if every surjective homomorphism is an isomorphism. Clearly every finite group is hopfian. that is not hopfian. by and and extend it to homomorphically. Note that is well-defined because. We also have and so is surjective. To prove that is not an isomorphism, let Then but.

Abstract. AbstractA module is called generalized Hopfian (gH) if any of its surjective endomorphisms has a small kernel. Such modules are in a sense dual to weakly co-Hopfian modules that were defined and extensively studied in [A. Haghany, M.R. Vedadi, J. Algebra () –].Author: A.

Ghorbani and A. Haghany.CRITICAL GROUPS FOR HOPF ALGEBRA MODULES 3 For an A-module V, if [V: S i] denotes the multiplicity of S i as a composition factor ofV, then () [V: S i] =dimHom A(P i,V). There are two Grothendieck groups, G0(A) and K0(A): • Theﬁrstone,G0(A),isdeﬁnedasthequotient ofthefree abelian group onthesetofallisomorphism classes [V] of A-modules V, subject to the relations [U] − [V] +[W] for.Books to Borrow.

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Full text of "3-manifold groups".