Some transitivity-like concepts in Abelian groups

Abstract

The classical notions of transitivity and full transitivity in Abelian p-groups have natural extensions to concepts called Krylov and weak transitivity.The interconnections between these four types of transitivity are determined for Abelian p-groups; there is a marked difference in the relationships when the prime p is equal to 2. In the final section the relationship between full and Krylov transitivity is examined in the case of mixed Abelian groups which are p-local in the sense that multiplication by an integer relatively prime to p is an automorphism.

On Transitivity-like Properties for Torsion-Free Abelian Groups

“…In regard to the last statement, which completely settles [17,Problem 44], it is worth to notice that in [1] was constructed a Krylov transitive Abelian 2-group which is neither fully transitive nor transitive. So, the last statement could be viewed as a torsion-free analogue of the cited construction.…”

“…This would imply that in H/5H ∼ = Z 5 we have that the element [2] is equal to either [1] or [−1] = [4], which is evidently not the case.…”

“…Proof. In virtue of Proposition 2.13 (1), it suffices to consider the case of finite m = m. To this purpose, let we have…”

“…Both notions of transitivity and full transitivity in Definition 1.1 were extended by Krylov in [16] to the following concept (note that Krylov used another terminology, whereas the term "Krylov transitive" was utilized for a first time by Goldsmith-Strüngmann -see, e.g., [1]): Definition 1.2. A group G is called Krylov transitive if, for any elements x, y ∈ G with χ G (x) = χ G (y), there exists an endomorphism of G mapping x to y.…”

“…The main motivation in writing up the present article is to initiate the comprehensive study of the existing relationships between all versions of the transitive properties of torsion-free Abelian groups, which examination was started in [3], and to compare them with the well-studied p-torsion case, where p is an arbitrary but fixed prime (for the latter case we refer the interested readers to [1], [4], [7] and [13], respectively). As it will be demonstrated in the sequel, the situation here is quite more difficult and complicated than it could be anticipated as well as there is no a satisfactory symmetry with the results obtained in the primary case.…”