Vol. 16, No. 1, 1966

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Quasi-isomorphism for infinite Abelian p-groups

Doyle Otis Cutler

Vol. 16 (1966), No. 1, 25–45

This paper is concerned with the investigation of two closely related questions. The first question is: What relationships exist between G and nG where G is an Abelian group and n is a positive integer?

It is shown that if Gand Hare Abelian groups, n is a positive integer and nGnH, then GH where G= S G and H= T H such that S and T are maximal n-bounded summands of Gand H, respectively. A corollary of this is: Every automorphism of nG can be extended to an automorphism of G.

We define two primary Abelian groups G and H to be quasi-isomorphic if and only if there exists positive integers m and n and subgroups S and T of G and H, respectively, such that pnG S, pmH T and ST, the second question is: What does quasi-isomorphism have to say about primary Abelian groups? It is shown that if two Abelian p-groups G and H are quasi-isomorphic then G is a direct sum of cyclic groups if and only if H is a direct sum of cyclic groups, G is closed if and only if H is closed, and G is a Σ-group if and only if H is a Σ-group.

Mathematical Subject Classification
Primary: 20.30
Received: 24 July 1964
Published: 1 January 1966
Doyle Otis Cutler