Vol. 16, No. 2, 1966

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First and second category Abelian groups with the n-adic topology

Edgar J. Howard

Vol. 16 (1966), No. 2, 323–329

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.

Mathematical Subject Classification
Primary: 20.30
Received: 22 September 1964
Published: 1 February 1966
Edgar J. Howard