Fully invariant subgroups, full transitivity, and homomorphism groups of Abelian groups.
type: Item
instance of: scholarly article
title: Fully invariant subgroups, full transitivity, and homomorphism groups of Abelian groups.
zbMATH Open document ID: 1131.20041
author: S. Ya. Grinshpon
author: P. A. Krylov
published in: Journal of Mathematical Sciences (New York)
publication date: 2007-08-07T00:00:00Z
review text: The paper contains an elaborate and detailed account of fully invariant subgroups in Abelian groups, fully transitive groups, and certain aspects of homomorphism groups. The bibliography comprises 71 items and in particular contains many papers published in Russian. There are certain properties of elements in an Abelian group that are necessarily preserved by homomorphisms. If \(\varphi\) is a homomorphism of the group \(G\), and \(x\in G\), then \(\mathbb{H}(x)\leq\mathbb{H}(\varphi(x))\) where \(\mathbb{H}\) is the height matrix. A group \(G\) is `fully transitive' if for any two elements \(x,y\in G\) with \(\mathbb{H}(x)\leq\mathbb{H}(y)\) there is an endomorphism \(\varphi\) of \(G\) such that \(\varphi(x)=y\). A related concept is that of an \(\mathbb{H}\)-group \(G\) that says that every fully invariant subgroup is of the form \(G(\mathbb{H})=\{x\in G:\mathbb{H}(x)\geq\mathbb{H}\}\). Every \(\mathbb{H}\)-group is fully invariant but there are fully invariant groups that are not \(\mathbb{H}\)-groups. These concepts have appeared in various classes of groups independently and in \(p\)-primary groups the height matrix appeared as height or Ulm sequence, in torsion-free groups as characteristic or type. The authors give a unified axiomatic treatment of these phenomena also dealing with properties of the lattice of fully invariant subgroups. They also introduce `fully transitive group families' \(\{A_i:i \in I\}\) that have the property that there is \(\varphi\in\Hom(A_i,A_j)\) with \(\varphi(a)=b\) whenever \(a\in A_i\), \(b\in A_j\) with \(\mathbb{H}(a)\leq\mathbb{H}(b)\). This concept is needed for a discussion of when direct sums and direct products of fully transitive groups are again fully transitive. Actually more general constructs, the \(\mathbf K\)-direct sums of families of groups are considered and a technical property, called `\(\mathbf K\)-monotonicity' is introduced. A sample result is Theorem 3.5. Let \(\{A_i\}_{i\in I}\) be a fully transitive grou…
reviewed by: Adolf Mader
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 20K27
Mathematics Subject Classification ID: 20K25
Mathematics Subject Classification ID: 20-02
Mathematics Subject Classification ID: 20E15
zbMATH DE Number: 5177686
zbMATH Keywords: automorphisms
zbMATH Keywords: endomorphisms
zbMATH Keywords: fully invariant subgroups
zbMATH Keywords: fully transitive Abelian groups
zbMATH Keywords: direct sums
zbMATH Keywords: direct products
zbMATH Keywords: height matrices
zbMATH Keywords: \(H\)-groups
zbMATH Keywords: lattices of fully invariant subgroups
MaRDI profile type: Publication
full work available at URL: s10958-005-0245-5
OpenAlex ID: W2089876411
cites work: FLATNESS AND THE RING OF QUASI-ENDOMORPHISMS
cites work: The flat dimension of mixed abelian groups as \(E\)-modules
cites work: Strongly Homogeneous Torsion Free Abelian Groups of Finite Rank
cites work: Finite rank torsion free Abelian groups and rings
cites work: Quasi-pure projective and injective torsion free Abelian groups of rank 2
cites work: Types of Elements and Characteristic Subgroups of Abelian Groups
cites work: The structure of large subgroups of primary abelian groups
cites work: Q4103735
cites work: Distributive modules
cites work: Q3033980
cites work: Q5639812
cites work: A sufficient condition for separability
cites work: THE INDEPENDENCE OF KAPLANSKY'S NOTIONS OF TRANSITIVITY AND FULL TRANSITIVITY
cites work: ON ENDOMORPHISM RINGS OF PRIMARY ABELIAN GROUPS, II
cites work: Q3754818
cites work: The homological dimension of an Abelian group as a module over its ring of endomorphisms. II
cites work: Q5682108
cites work: Q5688707
cites work: Transitive and fully transitive groups
cites work: Q5609457
cites work: Q5673086
cites work: A generalization of separable torsion-free Abelian groups
cites work: Q4948323
cites work: Q4428069
cites work: E-transitive torsion-free Abelian groups
cites work: On Transitive and Fully Transitive Primary Groups
cites work: Q5829819
cites work: Strongly homogeneous torsion-free Abelian groups
cites work: Q4761805
cites work: Endomorphism rings of Abelian groups
cites work: Q5639910
cites work: Q3918304
cites work: Q4897391
cites work: Q5668988
cites work: Q4052357
cites work: On -separable abelian groups
cites work: Q4156568
cites work: Primary abelian groups as modules over their endomorphism rings
cites work: Q5644070
cites work: Q3966350
cites work: Q5626713
cites work: Modules Whose Lattice of Submodules is Distributive
cites work: Distributive rings and endodistributive modules
cites work: Q3474807
cites work: Modules whose lattice of submodules is distributive
cites work: Q3474808
cites work: Rings of endomorphisms and distributivity
cites work: Q4219112
cites work: Q4953966
cites work: C\(_{\gamma}\)-groups and \(\gamma\)-basic subgroups
DOI: 10.1007/S10958-005-0245-5