Torsion-free abelian group

About: Torsion-free abelian group is a research topic. Over the lifetime, 71 publications have been published within this topic receiving 3099 citations.

A torsion-free abelian group exists whose quotient group modulo the square subgroup is not a \(nil\)-group

Abstract: The first example of a torsion-free abelian group such that the quotient group of modulo the square subgroup is not a nil-group is indicated (for both associative and general rings). In particular, the answer to the question posed by Stratton and Webb [‘Abelian groups, nil modulo a subgroup, need not have nil quotient group’, Publ. Math. Debrecen 27 (1980), 127–130] is given for torsion-free groups. A new method of constructing indecomposable nil-groups of any rank from to is presented. Ring multiplications on -pure subgroups of the additive group of the ring of -adic integers are investigated using only elementary methods.

A new filtration of the Magnus kernel of the Torelli group

Abstract: For a oriented genus g surface with one boundary component, S, the Torelli group is the group of orientation preserving homeomorphisms of S that induce the identity on homology. The Magnus representation of the Torelli group represents the action on F/F" where F=pi_1(S) and F" is the second term of the derived series. We show that the kernel of the Magnus representation, Mag(S), is highly non-trivial and has a rich structure as a group. Specifically, we define an infinite filtration of Mag(S) by subgroups, called the higher order Magnus subgroups, M_k(S). We develop methods for generating nontrivial mapping classes in M_k(S) for all k and g>1. We show that for each k the quotient M_k(S)/M_{k+1}(S) contains a subgroup isomorphic to a lower central series quotient of free groups E(g-1)_k/E(g-1)_{k+1}. Finally We show that for g>2 the quotient M_k(S)/M_{k+1}(S) surjects onto an infinite rank torsion free abelian group. To do this, we define a Johnson-type homomorphism on each higher order Magnus subgroup quotient and show it has a highly non-trivial image.

Torsion-free weakly transitive E-Engel Abelian groups

Abstract: It is proved that if all the endomorphisms of a reduced torsion-free weakly transitive Abelian group are bounded right-nilpotent, then its ring of endomorphisms is commutative. The ring of endomorphisms of a torsion-free Abelian group with periodic group of automorphisms and Engel ring of endomorphisms is also commutative.

The homological dimension of a torsion-free abelian group of finite rank as a module over its ring of endomorphisms

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