Can Aronszajn lines or Suslin lines ever be the orderings of abelian groups?

A torsion-free abelian group A is orderable, i.e. there is a linear order < on A such that if g < h then g + k < h + k for all g, h, k in A. The question is whether Suslin lines or Aronszajn lines can ever be orderings of abelian groups. See http://mathoverflow.net/questions/179618/can-suslin-lines-ever-be-orderings-of-abelian-groups