Subgroups of the Multiplicative Group of a Division Ring

Introduction.The general question of what groups can be embedded in the multiplicative group of a division ring is an unsolved problem, but some parts of the problem have been completely solved.It is well known that an abelian group can be embedded in a division ring if and only if its torsion subgroup is locally cyclic (see [4]).B. H. Neumann (see [10]) and A. I. Malcev (see [9]) showed that all ordered groups can be embedded in a division ring and Amitsur solved the problem for finite groups (see [2]).These results exhaust the major work that has been done on the problem to date.Iwasawa showed that any torsion-free nilpotent group can be ordered (see [7]) ; therefore, by Neumann's and Malcev's result any torsion-free nilpotent group can be embedded in a division ring.This paper will extend this result by showing that certain torsion-free solvable groups can be embedded in a division ring.This paper will also extend the results on abelian, finite and ordered groups by giving an embedding theorem for a group G which has an abelian, finite or ordered group H as a normal subgroup and a certain normal structure between Hand G.An embedding theorem.Let T denote a fixed division ring, x an indeterminate over T and 0 an automorphism of T. T(x ; 0> will denote the Hubert division ring of all formal power series in x over T (see [9]) r = { 2Z'¡cLnaix'¡ in T and n any integer}, where addition and equality are defined as in a commutative power series ring and multiplication is determined by the equation x ■ a = 0(a) • x for all a in T. Let T[x ; 0] denote the ring of polynomials in r , that is the ring generated by T and x; and let T(x; 0) denote the division ring generated by T and x.T[x; 0] is an Ore polynomial ring (see [11]) and thus has a quotient field which is uniquely determined up to isomorphism.Therefore T(x; 0) = {f(x)g-1(x) |/f», g(x) s* 0 in T[x; 0]} = {g~)f(x) |/(x), g(x) ± 0 in T[x; 0]}.This structure of T(x; 0) is of importance.