TORSION FREE LCA GROUPS, GROUPS WITH UNIQUE ROOTS AND A QUESTION OF A. G. MYASNIKOV: (II) CSA GROUPS

LCA groups were defined in [Torsion free LCS groups, groups with unique roots and a question of A. G. Myasnikov: (I) CT groups]. In that paper it was shown that a torsion free LCA group is CT if and only if it is a U-group. A stronger condition than commutative transitivity (CT) is the CSA property, namely, maximal abelian subgroups are malnormal. It is known that a sufficient (but not necessary) condition for a group to be CSA is that the centralizer of every nontrivial element be infinite cyclic. That raises the question of whether or not a group G has to be CSA if the centralizer of every nontrivial element is a torsion free locally cyclic group. Equivalently, must every LCA U-group be CSA? In this paper, we show that if is an integer, then the Baumslag-Solitar group is an LCA U-group which is not CSA.