Vol. 33, No. 3, 1970

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Ulm’s theorem for Abelian groups modulo bounded groups

Neal Hart

Vol. 33 (1970), No. 3, 635–640
Abstract

Let  be the category of Abelian groups, B the class of bounded Abelian groups. It is shown that if G and H are totally projective p-groups, then GH in the quotient category ÂB if and only if there exists an integer k 0 such that for all ordinals α and all integers r 0,

r∑+k           r∑+2k             r+∑k            r+∑2k
fG(α + j) ≦    fH(α +j) and    fH(α +j) ≦    fG (α + j).
j=k           j=0              j=k            j=0

This extends a similar result of R. J. Ensey for direct sums of countable reduced p-groups. It is also noted that if G and H are totally projective p-groups, then G is quasi-isomorphic to H if and only if there exists an integer k 0 such that for all integers n 0 and r 0,

r+k          r+2k
∑  f (n+ j) ≦ ∑  f  (n + j)
j=k  G         j=0 H

and

r∑+k           r+∑2k
fH(n + j) ≦     fG(n+ j), and fG(α) = fH (α )
j=k           j=0

for all α ω. This extends a similar result of R. S. Pierce and R. A. Beaumont for direct sums of countable reduced p-groups.

Mathematical Subject Classification
Primary: 20.30
Milestones
Received: 4 December 1969
Published: 1 June 1970
Authors
Neal Hart