The purpose of this note is
to give a generalization of the representation Theorems 33.1 and 33.2 of [2]. Let G be
an arbitrary abelian group and B = [⊕λ∖∈Λ⟨xλ⟩] ⊕ [⊕i≧1Bλ] be a p-basic
subgroup of G, cf. [3], where ⊕λ∈Λ⟨xλ⟩ is the torsionfree part. For all λ ∈ Λ let
(Fp∗)λ be a copy of the group of p-adic integers, and let (Fp)λ denote the
infinite cyclic group of finite p-adic integers in (Fp∗)λ. Then G can be mapped
homomorphically into the complete direct sum [⊕λ∈Λ∗(Fp∗)λ] ⊕ [⊕i≧1∗Bi] with
kernel pωG. Furthermore, the image of G is a p-pure subgroup which contains
[⊕λ∈Λ(Fp)λ] ⊕ [⊕i≧1Bi] as a p-basic subgroup and is in turn contained in
the p-adic completion of this subgroup (See Section 1 for definitions). This
representation is completely analogous to the representation theorem for
p-groups which is contained as a special case, and hopefully it is of similar
use.
