Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Intersections of pure subgroups in abelian groups

Authors: D. Boyer and K. M. Rangaswamy
Journal: Proc. Amer. Math. Soc. 81 (1981), 178-180
MSC: Primary 20K27
MathSciNet review: 593451
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that a subgroup $ H$ of an abelian group $ G$ is an intersection of pure subgroups of $ G$ if and only if, for all primes $ p$ and positive integers $ n$, $ {p^n}g \in H$ and $ {p^{n - 1}}g \notin H$ imply that there exists $ z \in G$ such that $ {p^n}z = 0$ and $ {p^{n - 1}}z \notin H$. This solves a problem posed by $ L$. Fuchs in [2] and [3].

References [Enhancements On Off] (What's this?)

  • [1] Bernard Charles, Une caractérisation des intersections de sous-groupes divisibles, C. R. Acad. Sci. Paris 250 (1960), 256–257 (French). MR 0109176
  • [2] L. Fuchs, Review of [1], #64, Math. Reviews 22 (1961), 10-11.
  • [3] L. Fuchs, Recent results and problems on abelian groups, Topics in Abelian Groups (Proc. Sympos., New Mexico State Univ., 1962), Scott, Foresman and Co., Chicago, Ill., 1963, pp. 9–40. MR 0169906
  • [4] -, Infinite abelian groups. I, Academic Press, New York, 1970.
  • [5] Paul Hill and Charles Megibben, Minimal pure subgroups in primary groups, Bull. Soc. Math. France 92 (1964), 251–257. MR 0186731
  • [6] Samir A. Khabbaz, The subgroups of a divisible group 𝐺 which can be represented as intersections of divisible subgroups of 𝐺, Pacific J. Math. 11 (1961), 267–273. MR 0122871
  • [7] Charles Megibben, On subgroups of primary abelian groups, Publ. Math. Debrecen 12 (1965), 293–294. MR 0186732
  • [8] K. M. Rangaswamy, Characterisation of intersections of neat subgroups of abelian groups, J. Indian Math. Soc. (N.S.) 29 (1965), 31–36. MR 0183778

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 20K27

Retrieve articles in all journals with MSC: 20K27

Additional Information

Keywords: Abelian group, pure subgroup
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society