Fully invariant subgroups, full transitivity, and homomorphism groups of Abelian groups.
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 group family with \(\mathbf K\)-monotonicity. Then \(A=\bigoplus_{\mathbf K}A_i\) is fully transitive. In the second chapter \(\mathbb{H}\)-groups are considered and the question how and when this property is inherited by fully invariant subgroups and by generalized direct sums of families of groups. Here a concept appears that is variously called ``contract property'' and ``contrast property''. In particular, splitting mixed groups are studied in view of being \(\mathbb{H}\)-groups. As a sample result there is Theorem 12.1. Let \(A=T\oplus G\) be a splitting mixed group. Then \(A\) is an \(\mathbb{H}\)-group if and only if the torsion-free part \(G\) is an \(\mathbb{H}\)-group, every \(p\)-part \(T_p\) is fully transitive, and \(G\) is \(p\)-divisible if \(T_p\) is unbounded. In the third chapter properties of the lattice of all fully invariant subgroups, such as distributivity and ``generalized'' distributivity are studied for torsion-free groups and \(p\)-groups. Corollary 15.10. states that the lattice of wide subgroups of an Abelian \(p\)-group is complete if and only if the reduced part of the group is bounded. -- Here Pierce's ``large subgroups'' morphed into ``wide subgroups'' via the Russian language. Another nice result is Corollary 16.9. where \(\Omega(G)\) is the set of types of rank-one direct summands: If \(G\) and \(G'\) are two separable torsion-free groups with \(\Omega(G)=\Omega(G')\), then the lattices of fully invariant subgroups of the groups are isomorphic. The final fourth chapter deals with \(\Hom(A,B)\) as a bimodule over the endomorphism rings \(\text{E}(A)\) and \(\text{E}(B)\). The main results determine when the \(\text{E}(A)\) module \(\Hom(A,B)\) is injective. The paper is carefully written, contains all necessary definitions and offers detailed arguments.
scientific article; zbMATH DE number 5177686
scientific article