Vol. 42, No. 1, 1972

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 331: 1
Vol. 330: 1  2
Vol. 329: 1  2
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Online Archive
Volume:
Issue:
     
The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Officers
 
Subscriptions
 
ISSN 1945-5844 (electronic)
ISSN 0030-8730 (print)
 
Special Issues
Author index
To appear
 
Other MSP journals
Primary groups whose subgroups of smaller cardinality are direct sums of cyclic groups

Paul Daniel Hill

Vol. 42 (1972), No. 1, 63–67
Abstract

Let G denote a primary abelian group. The conjecture (partially supported by a theorem of Nunke) that there exists, for each infinite cardinal m, a group G of cardinality m that is not a direct sum of cyclic groups but has the property that each subgroup of G having cardinality less than m is a direct sum of cyclic groups is shown to be false. More specifically, it is shown that if a primary group has cardinality ω and each subgroup of smaller cardinality is a direct sum of cyclic groups, then so is the group.

Further, we show that if G is the union of a countable chain of pure subgroups, then G is a direct sum of cyclic groups if and only if the subgroups in the given chain are direct sums of cyclic groups.

Mathematical Subject Classification 2000
Primary: 20K25
Milestones
Received: 17 February 1970
Revised: 14 December 1971
Published: 1 July 1972
Authors
Paul Daniel Hill