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Throughout this paper
the word group shall mean Abelian group. The n-adic topology of a group
G is formed by taking the subgroups k!G as a base for the neighborhood
system of the identity where k is a nonnegative integer. In this paper we
list some properties of first and second category groups with the n-adic
topology (a group is of first category if it is a countable union of nowhere dense
sets).
We characterize first and second category groups and prove the following:
Theorem: A torsion group is of second category if and only if G = H ⊕ D where
H is bounded and D is divisible.
Theorem: Every torsion homomorphic image of a second category (e.g. complete)
group is the direct sum of a bounded group and a divisible group.
Theorem: If G is reduced and of second category and G = ∑
Gα, then there exists
an integer n such that nGα = 0 for all but finitely many α.
Theorem: If T is torsion, T is isomorphic to the torsion subgroup of a second
category group.
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