Structure theorems for infinite abelian groups
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In this paper we have determined the structure of divisible groups, certain primary groups, and countable torsion groups. Chapter 1 introduces two important infinite abelian groups, R and Z(p^∞). The structure of these groups is completely known and we have given most of the important properties of these groups in Chapter 1. Of special importance is the fact that a divisible group can be decomposed into a direct sum of groups each isomorphic to R or Z(p^∞). This is Theorem 2.12 and it classifies all divisible groups in terms of these two well-known groups. Theorem 1.6 is of great importance since it reduces the study of torsion groups to that of primary groups. We now have that Theorems 3.3 and 5.5 apply to countable torsion groups as well as primary groups. Theorem 3.3 gives a necessary and sufficient condition for an infinite torsion group to be a direct sum of cyclic groups. These conditions are more complicated than the finite case. From Theorem 3.3, we derived Corollary 3.5. This result is used later on to get that the Ulm factors of a group are direct sums of cyclic groups. In essence, Ulm's theorem says that a countable reduced primary group can be determined by knowing its Ulm type and its Ulm sequence. Now by Corollary 3.5, we have only to look at the number of cyclic direct summands of order p^n (for all integers n) for each Ulm factor. This gives us a system of invariants which we can assign to the group. Once again, these invariants are much harder to arrive at than in the finite case.
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