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This paper is concerned
with the investigation of two closely related questions. The first question is: What
relationships exist between G and nG where G is an Abelian group and n is a
positive integer?
It is shown that if G′ and H′ are Abelian groups, n is a positive integer and
nG′≅nH′, then G≅H where G′ = S ⊕ G and H′ = T ⊕ H such that S and T are
maximal n-bounded summands of G′ and H′, respectively. A corollary of
this is: Every automorphism of nG can be extended to an automorphism of
G.
We define two primary Abelian groups G and H to be quasi-isomorphic if and
only if there exists positive integers m and n and subgroups S and T of G and H,
respectively, such that pnG ⊂ S, pmH ⊂ T and S≅T, the second question is: What
does quasi-isomorphism have to say about primary Abelian groups? It is shown that
if two Abelian p-groups G and H are quasi-isomorphic then G is a direct
sum of cyclic groups if and only if H is a direct sum of cyclic groups, G is
closed if and only if H is closed, and G is a Σ-group if and only if H is a
Σ-group.
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