Homomorphic stability of Abelian groups

In the paper, necessary and sufficient conditions for an Abelian group A to be isomorphic to the endomorphism group End(A) are obtained. The classes of periodic Abelian groups, divisible Abelian groups, nonreduced Abelian groups, and reduced algebraically compact Abelian groups are considered. For certain classes of Abelian groups, the isomorphism problem for a group and its endomorphism group is solved under the assumption that the endomorphism group itself has the corresponding property.

Let $$\Lambda$$ be a class of Abelian groups. A group $$A\in\Lambda$$ is said to be determined by its endomorphism semigroup $$E^\star(A)$$ in the class $$\Lambda$$ if every isomorphism $$E^\star(A)\cong E^\star(B)$$ , where $$B\in\Lambda$$ , implies the isomorphism $$A\cong B$$ . The paper describes those Abelian groups in the class $$\mathscr Q\mathscr D_{\mathrm{cd}}$$ of completely decomposable quotient divisible Abelian groups which are determined by their endomorphism semigroups in the class $$\mathscr Q\mathscr D_{\mathrm{cd}}$$ .

The fundamental theorem of abelian groups states that any finitely generated abelian group is a direct sum of cyclic groups. This theorem plays a fundamental role in the structure theory of abelian groups. It has fascinated many algebraists to look at this theorem from different points of view:- (a) To find suitable generalizations of this theorem for modules over certain classes of rings, e.g. Dedekind domain, hereditary noetherian prime rings, valuation rings etc. (b) To create suitable versions of this theorem in some module categories and use such versions to develop the structure theory of such module categories. For example, torsion abelian group-like modules, modules with finitely generated submodules direct sums of multiplication modules. (c) To find those rings for which certain versions of the fundamental theorem of abelian groups hold. For example rings over which all finitely generated modules are direct sums of cyclic modules. (d) To examine the structure of certain classes of abelian groups and to try to find modules with similar structures. (e) To find roles of answers to some of above types of questions in the general theory of modules.

It is proved that all cohomology groups of a group G acting on an abelian group of a certain type vanish if the action of some subnormal abelian subgroup of G is fixed-point-free.This result is then applied to obtain results about group extensions and about the complete reducibility of linear groups.Introduction.The purpose of this paper is to prove the Proposition below and its corollary, Theorem 1, and to derive from the latter some consequences about group extensions and about the complete reducibility of linear groups.These results are contained in Corollaries 2-4.Statement (a) of § 1 and part (b) of the Lemma, §2, are related to the surrounding material, but are not used in the subsequent proofs.I am indebted to several colleagues for useful suggestions in connection with this paper.I am also grateful to the referee for his valuable comments, especially for pointing out that the original version of Theorem 1 could be strengthened to its present form.In what follows, a Vx-module for a group G will mean a finite direct sum 2 V¡, where V¡ is a finite-dimensional vector space over a field k¡, and V¡ is a /c,G-module.Fory > 2, a VJ-module is a G-module A for which there is a short exact sequence 0 -> A' -> A -» A" -» 0, where A' is a K'-module and A" is a F-7-1-module.A is a W-module if it is a F^-module for somtj.It can be shown that every iV-module is a direct sum of a torsion group of finite exponent and a torsion free divisible group (i.e., a vector space over Q).Conversely, one can show that every such abelian group is a W-mod\i\c (for trivial group action).Moreover, if such a group A = T © F is given as a Gmodule, then A is automatically a W-module for G if the torsion part T is finite and Fis finite dimensional over Q.This follows by induction on \T\, the order of T, from the exact sequence 0 -> Sp -* A -> A/Sp = T/Sp © F '-* 0, wherep is any prime dividing |7j and Sp = [x G A: px = 0).(If T = 0, the statement is clear.)Proposition.Let G be a group, N a subnormal abelian subgroup of G, k a

A class of Abelian groups with hereditary rings of endomorphisms

If H is a subgroup of an abelian group G and φ ∈ End(G), H is called φ-inert (and φ is H-inertial) if φ(H) ∩ H has finite index in the image φ(H). The notion of φ-inert subgroup arose and was investigated in a relevant way in the study of the so called intrinsic entropy of an endomorphism φ, while inertial endo-morphisms (these are endomorphisms that are H-inertial for every subgroup H) were intensively studied by Rinauro and the first named author.A subgroup H of an abelian group G is said to be fully inert if it is φ-inert for every φ ∈ End(G). This property, inspired by the “dual” notion of inertial endomorphism, has been deeply investigated for many different types of groups G. It has been proved that in some cases all fully inert subgroups of an abelian group G are commensurable with a fully invariant subgroup of G (e.g., when G is free or a direct sum of cyclic p-groups). One can strengthen the notion of fully inert subgroup by defining H to be uniformly fully inert if there exists a positive integer n such that |(H + φH)/H| ≤ n for every φ ∈ End(G). The aim of this paper is to study the uniformly fully inert subgroups of abelian groups. A natural question arising in this investigation is whether such a subgroup is commensurable with a fully invariant subgroup. This paper provides a positive answer to this question for groups belonging to several classes of abelian groups.

The purpose of this survey is to provide an introduction to the interesting role which set-theoretic methods have begun to play in the problem of determining the structure and classification of certain classes of abelian groups, in particular, the class of ω1-separable groups, i.e., groups such that every countable subset is contained in a Σ -cyclic summand.

Let T be an abelian group and λ an uncountable regular cardinal. The question of whether there is a λ-universal group U among all torsion-free abelian groups G of cardinality less than or equal to λ satisfying Exta(G,T) = 0 is considered. U is said to be λ-universal for T if, whenever a torsion-free abelian group G of cardinality at most λ satisfies Ext(G,T)=0, there is an embedding of G into U. For large classes of abelian groups T and cardinals λ, it is shown that the answer is consistently no, that is to say, there is a model of ZFC in which, for pairs T and λ, there is no universal group. In particular, for T torsion, this solves a problem by Kulikov.

Using sets of finitely generated Abelian groups closed under the discrimination operator, we describe principal universal classes 𝒦 within a quasivariety 𝔄p, the class of groups whose periodic part is a p-group for a prime p. Also the concept of an algebraically closed group in 𝒦 is introduced, and such groups are classified.

The hopficity property in a well-known class of Abelian groups—completely decomposable torsion-free groups—as well as in a class of modules over generalized matrix rings is studied. Examples of non-Hopfian completely decomposable torsion-free group are constructed.

Epimorphic images of compact (algebraically compact) abelian groups are called cotorsion groups after Harrison. In a recent paper, Ph. Schultz raised the question whether “cotorsion” is a property which can be recognized by its small cotorsion epimorphic images: If G is a torsion-free group such that every torsion-free reduced homomorphic image of cardinality is cotorsion, is G necessarily cortorsion? In this note we will give some counterexamples to this problem. In fact, there is no cardinal k which is large enough to test cotorsion.

It is shown that each commutative Artin local ring having each of its ideals generated by two elements is the homomorphic image of a one-dimensional local complete intersection ring which also has each of its ideals generated by two elements. It is indicated how this can be applied to show that the property that each ideal is projective over its endomorphism ring does not pass to homomorphic images, and in determining the commutative group rings with the two-generator property.

There is an equivalence relation on the set of smooth maps of a manifold into the stable unitary group, defined using a Chern-Simons type form, whose equivalence classes form an abelian group under ordinary block sum of matrices. This construction is functorial, and defines a differential extension of odd K-theory, fitting into natural commutative diagrams and exact sequences involving K-theory and differential forms. To prove this we obtain along the way several results concerning even and odd Chern and Chern-Simons forms.

Let be a commutative ring with identity. We consider commutative -algebras generated by idempotents over . For an algebraically closed field and a -algebra , we show that is generated by idempotents over if and only if is a homomorphic image of the group algebra of an abelian torsion group having no element whose order is equal to the characteristic of . We also give a necessary and sufficient condition for an -algebra generated by a finite number of idempotents over to be a homomorphic image of a group algebra over .

Let B B be a subgroup of an abelian group G G such that G / B G/B is isomorphic to a direct sum of copies of an abelian group A A . For B B to be a direct summand of G G , it is necessary that G G be generated by B B and all homomorphic images of A A in G G . However, if the functor Hom ⁡ ( A , − ) \operatorname {Hom} (A, - ) preserves direct sums of copies of A A , then this condition is sufficient too if and only if M ⊗ E ( A ) A M{ \otimes _{E(A)}}A is nonzero for all nonzero right E ( A ) E(A) -modules M M . Several examples and related results are given.