ABSTRACT
Chapter 23 Quotient Divisible Groups, ω-Groups, and an Example of Fuchs James D. Reid Department of Mathematics and Computer Science, Wesleyan University, Middletown, Connecticut 06459, USA jreid@wesleyan.edu
23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 23.2 On ω-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 23.3 Three Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 23.4 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 23.5 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 23.6 Endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
23.1 Introduction Groups here will generally be torsion free and always abelian. In a classical and seminal paper [2] Beaumont and Pierce studied, among other things, what they called quotient divisible groups. These are the torsion free groups (for them of finite rank) that contain a free subgroup modulo which they are divisible. Without loss of generality the free subgroup can be taken to be full, that is, with a torsion quotient, so the group G in question fits into an exact sequence of the form
0 → X → G → D → 0
with X free and D torsion divisible. These groups are of interest because, for example, the additive groups of full subrings of semisimple rational algebras have such structure.
