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TORSION FREE LCA GROUPS, GROUPS WITH UNIQUE ROOTS AND A QUESTION OF A. G. MYASNIKOV: (I) CT GROUPS
Keywords:
LCA group, commutative transitive, U-group, discriminates, CT groupsDOI:
https://doi.org/10.17654/0972087126029Abstract
Let us say a group G is LCA provided every abelian subgroup of G is locally cyclic. A. G. Myasnikov posed the question of whether or not every torsion free LCA group must be commutative transitive in the sense that the relation of commutativity is transitive on the nonidentity elements. Pavel Shumyatsky observed that a counterexample to Myasnikov’s question is contained in a classical result of Adyan. This paper is meant to be expository and light reading. None the less, we do prove one theorem, namely, a torsion free LCA group G is commutative transitive if and only if roots in G (when they exist) are unique.
Received: August 17, 2025
Accepted: November 19, 2025
References
[1] S. I. Adyan, On some torsion-free groups, Izv. Akad. Nauk. SSS Ser. MAT 35 (1971), 459-468.
[2] G. Baumslag, Lecture Notes on Nilpotent Groups, American Math. Soc., Providence, RI, 1969.
[3] A. M. Gaglione, S. Lipschutz and D. Spellman, Some properties of nilpotent groups, Algebra and Discrete Math. 4 (2009), 66-77.
[4] N. Harrison, Real length functions in groups, Trans. Amer. Math. Soc. 174 (1972), 77-106.
[5] W. Hodges, Building Models by Games, Dover, NY, 2006.
[6] A. G. Kurosh, The Theory of Groups, Chelsea, NY, 1960.
[7] H. Neumann, Varieties of Groups, Springer-Verlag, NY, 1967.
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