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On Purifiable Subsocles of a Primary Abelian Group

Published online by Cambridge University Press:  20 November 2018

John Irwin  and
James Swanek
John Irwin
Affiliation:
Wayne State University, Detroit, Michigan
James Swanek
Affiliation:
Ford Motor Company, Dearborn, Michigan
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In this paper we shall investigate an interesting connection between the structure of G/S and G, where S is a purifiable subsocle of G. The results are interesting in the light of a counterexample by Dieudonné [3, p. 142] who exhibits a primary abelian group G, where G/S is a direct sum of cyclic groups, but G is not a direct sum of cyclic groups. Surprisingly, the assumption of the purifiability of S allows G to inherit the structure of G/S. In particular, we show that if G/S is a direct sum of cyclic groups and S supports a pure subgroup H, then G is a direct sum of cyclic groups and if is a direct summand of G which is of course a direct sum of cyclic groups. It is also shown that if G/S is a direct sum of torsion-complete groups and S supports a pure subgroup H, then G is a direct sum of torsion-complete groups and H is a direct summand of G, and is also a direct sum of torsion-complete groups.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 1 , February 1971 , pp. 48 - 57
Copyright
Copyright © Canadian Mathematical Society 1971

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