Affine Completeness and Lexicographic Product Decompositions of Abelian Lattice Ordered Groups

In this paper it is proved that an abelian lattice ordered group which can be expressed as a nontrivial lexicographic product is never affine complete. This work was supported by VEGA grant 2/1131/21.     

Coxeter群的半直积分解(英文)

本文证明了满足一定条件的Coxeter群都可以分解成两个Coxeter群的半直积.     

Double Coset Decompositions of Groups

We show that residually finite or word hyperbolic groups which can be decomposed as a finite union of double cosets of a cyclic subgroup are necessarily virtually cyclic, and we apply this result to the study of Frobenius permutation groups. We show that in general, finite double coset decompositions of a group can be interpreted as an obstruction to splitting a group as a free product with amalgamation or an HNN extension.     

Sequences of Endomorphism Groups of Abelian Groups

In the paper, Problem 18.3 of the book “Abelian groups” (2015) by L. Fuchs is solved in the case of Abelian groups with finite p-ranks. For an Abelian group A, a sequence of groups (A_n) is considered, where A₀ = A and A_n+1 = End A_n. It is shown that, if all p-ranks of the group A are finite, then this sequence can stabilize either after A₀ or after A₁.

     

Characterizing Automorphism Groups of Ordered Abelian Groups

A proof is given of the following theorem, which characterizesfull automorphism groups of ordered abelian groups: a groupH is the automorphism group of some ordered abelian group ifand only if H is right-orderable. 2000 Mathematics Subject Classification20K15, 20K20, 20F60, 20K30 (primary); 03E05 (secondary).     

A-Solvability and Mixed Abelian Groups

An abelian group A is an S-group (S ⁺-group) if every subgroup B ≤ A of finite index is A-generated (A-solvable). This article discusses some of the differences between torsion-free S-groups and mixed S-groups, and studies (mixed) S- and S ⁺-groups, which are self-small and have finite torsion-free rank.     

Abelian Groups and Regular Modules

The structure of the additive group of a regular module is considered. Abelian groups that are regular modules over their rings of endomorphisms are studied. Nonreduced endoregular groups and endoregular torsion-free groups of finite rank are described.     

Local Abelian Torsion-Free Groups

A representation for p-local Abelian torsion-free groups of finite rank is obtained in terms of homomorphisms of p-adic modules of finite rank with fixed basis.     

Homomorphically Stable Abelian Groups

Mathematical Notes - A group is said to be homomorphically stable with respect to another group if the union of the homomorphic images of the first group in the second group is a subgroup of the...     

On Abelian Groups Close to E-Solvable Groups

E-nilpotent and E-solvable Abelian groups are studied. The properties of such groups are studied, and examples illustrating the differences and connections between the investigated classes of groups are presented. The structure of E-solvable periodical, completely decomposable, coperiodic, split mixed, and other groups is shown.     

Absolute Nil-Ideals of Abelian Groups

It is known that in an Abelian group G that contains no nonzero divisible torsion-free subgroups the intersection of upper nil-radicals of all the rings on G is $\bigcap\limits_{p} pT(G)$ , where T(G) is the torsion part of G. In this work, we define a pure fully invariant subgroup G*???T(G) of an arbitrary Abelian mixed group G and prove that if G contains no nonzero torsion-free subgroups, then the subgroup $\bigcap\limits_{p} pG^{*}$ is a nil-ideal in any ring on G, and the first Ulm subgroup G1 is its nilpotent ideal.