The results in this note arose from considering the question: What are the abelian subgroups of a one-relator group? The additive group of p-adic rationals and the free abelian group of rank 2 are certainly subgroups of a one-relator group. For example in
In a survey article [1] Baumslag posed the problem of determining the abelian subgroups of a one-relator group. The solution of this problem was stated but not proved in [5], and partly solved by Moldavanskii [4]. In this paper it will be proved that the centralizer of every non-trivial element in a one-relator group with torsion is cyclic, and that the soluble subgroups of a one-relator groups with torsion are cyclic groups or the infinite dihedral group. That both types of groups may occur as subgroups is easily seen by considering
Abstract. Let K = Q( √ −3) or Q( √ −1) and let Cn denote the cyclic group of order n. We study how the torsion part of an elliptic curve over K grows in a quadratic extension of K. In the case E(K)[2] ≈ C2 ⊕ C2 we determine how a given torsion structure can grow in a quadratic extension and the maximum number of quadratic extensions in which it grows. We also classify the torsion structures which occur as the quadratic twist of a given torsion structure.
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