Vol. 24, No. 1, 1968

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Isomorphism invariants for Abelian groups modulo bounded groups

Ronald J. Ensey

Vol. 24 (1968), No. 1, 71–91

Let 𝒜 be the category of Abelian groups, let be the class of bounded Abelian groups, and form the quotient category 𝒜. The principal goal of this paper is a complete set of invariants for direct sums of countable reduced p-groups, such groups considered as objects of the category 𝒜. Specifically, it will be shown that two direct sums of countable reduced p-groups G and H are isomorphic in 𝒜if and only if there is an integer k 0 such that for all ordinal numbers α and all integers r 0

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


∑r               r+∑2k
fB(α + k+ j) ≦    fG(α + j)
j=0               j=0

where fG(β) and f1J(β) denote the β-th Ulm invariants of G and H, respectively. Thus a complete set of 𝒜-isomorphism invariants for such groups is an equivalence class of Ulm invariants, the equivalence relation being given by these two inequalities.

Mathematical Subject Classification
Primary: 20.30
Received: 13 December 1966
Published: 1 January 1968
Ronald J. Ensey