| Summary: | 碩士 === 淡江大學 === 數學系 === 82 === The notion of pure subgruups is intermediate between sub-
groups and direct summands. It is well-known that every
direct summand is a pure subgroup. But it is not necessary
that pure subgroups are direct summands. Therefore, we want
to discuss such groups which have the property: if H is a pure
subgroup of G, then H must be a direct summand of G. These
groups possess a number of remarkable property which are
characteristic for them. In 1954, Kaplansky defined the
algebraically compact groups in his books "Infinite Abelian
Groups". He has treated algebra- ically compact groups in a
topological way. By taking topo- logical approach, Fuchs
has discuss the structure of these groups. Then, Legg
and Walker tried to describe these groups non-topological
discussion of structure theory. Thus, they use algebraic
treatment in their paper. In this paper, we will discuss
these groups by algebraic treatment. But the definition of
algebraically compact groups is different. Thus, some
theorems are proved by using this definition. In Theorem 2
and Theorem 6, it will be proved that these definitions are
equivalent. The only example shows us two facts: pure
subgroup is not a direct summand and the direct sum of
algebraically compact groups is not algebraically compact. The
most important theorem is Theorem 7. Further, in the
conclusion, using the structure theorem we propose the
following conjecture: (1) A torsion group is algebraically
compact if and only if it is the direct sum of a bounded
group and a divisible group. (This is true.) (2) A torsion
free group is algebraically compact if and only if it is
divisible.(This may be false.)
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