Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-08T02:43:47.075Z Has data issue: false hasContentIssue false
  • English
  • Français

Criteria for Total Projectivity

Published online by Cambridge University Press:  20 November 2018

Paul Hill
Paul Hill*
Affiliation:
Auburn University, Auburn, Alabama
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

All groups herein are assumed to be abelian. It was not until the 1940's that it was known that a subgroup of an infinite direct sum of finite cyclic groups is again a direct sum of cyclics. This result rests on a general criterion due to Kulikov [7] for a primary abelian group to be a direct sum of cyclic groups. If G is p-primary, Kulikov's criterion presupposes that G has no elements (other than zero) having infinite p-height. For such a group G, the criterion is simply that G be the union of an ascending sequence of subgroups Hn where the heights of the elements of Hn computed in G are bounded by some positive integer λ(n). The theory of abelian groups has now developed to the point that totally projective groups currently play much the same role, at least in the theory of torsion groups, that direct sums of cyclic groups and countable groups played in combination prior to the discovery of totally projective groups and their structure beginning with a paper by R. Nunke [11] in 1967.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 4 , 01 August 1981 , pp. 817 - 825
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Fuchs, L., Infinité abelian groups, vol. II (Academic Press, New York and London, 1973).Google Scholar
2. Griffith, P., Infinite abelian group theory (The University of Chicago Press, Chicago and London, 1970).Google Scholar
3. Hill, P., Isotype subgroups of direct sums of countable groups, Illinois J. Math. 18 (1969), 281–290.Google Scholar
4. Hill, P., Primary groups whose subgroups of smaller cardinality are direct sums of cyclic groups, Pacific J. Math. 41 (1972), 63–67.Google Scholar
5. Hill, P., Countable unions of totally projective groups, Trans. Amer. Math. Soc. 190 (1974), 385–391.Google Scholar
6. Hill, P., The third axiom of countability for abelian groups, Proc. Amer. Math. Soc. (to appear).Google Scholar
7. Kulikov, L., On the theory of abelian groups of arbitrary cardinality (Russian), Math. Sb. 9 (1941), 165–182.Google Scholar
8. Linton, R. and Megibben, C., Extensions of totally projective groups, Proc. Amer. Math. Soc. 64 (1977), 35–38.Google Scholar
9. Megibben, C., The generalized Kulikov criterion, Can. J. Math. 21 (1969), 1192–1205.Google Scholar
10. Megibben, C., A generalization of the classical theory of primary groups, Töhoku Math. J. 22 (1970), 344–356.Google Scholar
11. Nunke, R., Homology and direct sums of abelian groups, Math. Z. 101 (1967), 182–212.Google Scholar