Vol. 22, No. 3, 1967

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On the structure of Tor. II

Ronald John Nunke

Vol. 22 (1967), No. 3, 453–464
Abstract

The following results are proved:

If A and B are abelian p-groups and the length of A is greater than the length of B, then Tor(A,B) is a direct sum of countable groups if and only if (i) B is a direct sum of countable groups and (ii) if the β-th Ulm invariant of B is not zero, then every pβA-high subgroup of A is a direct sum of countable groups.

If β is an ordinal, A is a p-group, and if one p6A-high subgroup of A is a direct sum of countable groups then every pβA-high subgroup of A is a direct sum of countable groups.

If A and B are p-groups of cardinality 1 without elements of infinite height, then Tor(A,B) is a direct sum of cyclic groups.

For each n with 1 n < ω, there is a p-group G without elements of infinite height such that G is not itself a direct sum of cyclic groups but every subgroup of G having cardinality n is a direct sum of cyclic groups.

Mathematical Subject Classification
Primary: 20.30
Milestones
Received: 8 June 1966
Published: 1 September 1967
Authors
Ronald John Nunke