Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-24T02:55:16.146Z Has data issue: false hasContentIssue false
  • English
  • Français

Vertical Subgroups of Primary Abelian Groups

Published online by Cambridge University Press:  20 November 2018

K. Benabdallah ,
B. Charles  and
A. Mader
K. Benabdallah
Affiliation:
Département de Mathématiques et de Statistique Université de Montréal C.P. 612 8, Succ. A Montréal, Québec H3C 3J7
B. Charles
Affiliation:
Institut de Mathématiques Université des Sciences et Techniques de Languedoc Montpellier, France
A. Mader
Affiliation:
2565 The Mail Department of Mathematics University ofHawaii Honolulu, Hawaii 96822 U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

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.

Motived by an intrinisic necessary condition for the purifiability of subgroups of primary abelian groups due to K. Benabdallah and T. Okuyama we introduce new functors on the category of pairs (G, A),where A is a subgroup of G,to the category of Z/pZ-vector spaces. The vanishing of these functors leads to the notion of vertical subgroup which is a weakening of purity but also an essential component of the latter. In fact, a vertical subgroup is pure if and only if it is neat. We establish various facts about vertical subgroups and “maximal” vertical subgroups and apply the resulting theory to the problem of purifiability. We show that the class of quasi-complete groups is precisely the class of reduced groups in which every subgroup satisfying the intrinsic necessary condition for purifiability is in fact purifiable. This is also the class of reduced p-groups in which the maximal vertical subgroups are precisely the pure subgroups.

Keywords

46L5546L40
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 1 , 01 February 1991 , pp. 3 - 18
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Benabdallah, K., Irwin, J., On minimal pure subgroups, Publ. Math. Debrecen 23(1973), 111–114.Google Scholar
2. Benabdallah, K., Mader, A., When is multiplication by p a closed map?, to appear, Proceedings of the International Conference on Algebra, Novosibirsk, USSR, August 1989.Google Scholar
3. Benabdallah, K., Okuyama, T., Onpurifiable subgroups of primary abelian groups, to appear, Communications in Algebra.Google Scholar
4. Charles, B., Etude sur les sous-groupes d'un groupe abélien, Bull. Soc. Math. France 88(1960), 217-227'.Google Scholar
5. Fuchs, L., Infinite Abelian Groups, Vol. I, II, Academic Press, 1970 and 1973.Google Scholar
6. Head, T., Remarks on a problem in primary abelian groups, Bull. Soc. Math. France 91(1963), 102–112.Google Scholar
7. Hill, P., Ch. Megibben, Minimal pure subgroups in primary groups, Bull, Soc. Math. France 92(1964), 251–257.Google Scholar
8. Hill, P., Quasi-closed primary groups, Acta Math. Acad. Sci. Hungar. 16(1965), 271–274.Google Scholar
9. Hill, P., On primary groups with countable basic subgroups, Trans. Amer. Math. Soc. 124(1966), 49–59.Google Scholar